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It’s tough to be the next great one. Grant Hill and Vince Carter never much cared for that Next Jordan label, and many others have tripped up being advertised as the next Wilt or Russell or Oscar or Magic or Bird.

Mike Dunleavy Jr. as well since being a good shooting, unselfish, ball handling white guy, well, you have to be the next Larry Bird.

Wednesday at the team’s Berto Center practice facility the newest Bull offered a tight smile when asked about who he was supposed to be as the No. 3 pick in the NBA draft off championship Duke U. team in 2002 and who he has become after 11 seasons in the NBA.

“I always try to run my own race and play the game the right way,” said Dunleavy, who signed a two-year contract with the Bulls on the first day of official free agent signings. “It’s tough to come in a young guy, a high pick, a young organization that is changing. There was a lack of continuity in Golden State. But I got to Indiana and found my way (averaging a career high 19.1 points in 2007-08). Some (knee) injuries knocked me off track, but if you told me I’d be going into my (12th) year playing for the Bulls I’d forget what happened in the last and take it.”

And the Bulls will as well as they targeted Dunleavy from the first moments of free agency July 1, contacting him and making a somewhat unexpected offer as the Bulls used their taxpayer midlevel exception of slightly more than $3 million per season. The Bulls, one of six teams in the luxury tax, weren’t expected to use the additional exception given the multiples once a team is in the penalty tax.

But seeing a need for perimeter shooting with the return of Derrick Rose, the Bulls, who moved to fourth overall in NBA payroll, got one of the better ones in Dunleavy (eighth in the league in threes last season) and a versatile player who is well regarded for both his passing and ball handling.

And perhaps as importantly, the Bulls signed a highly motivated player, an in-demand free agent who wanted to come to Chicago and turned down considerably more money and security.

“There were some excellent teams after him,” said his father, Mike Dunleavy Sr., the longtime NBA coach and player. “But he felt like the whole package of what the Bulls offered was what he wanted. We talked to probably three teams that were interested in him that could be in the final six in the NBA. He’s 32, with maybe three or four years left to play. One team offered a four-year deal. He was offered twice the money. But it’s not about the money. It’s about the ring. The better players he’s with the better he’ll be. He’s never even played with an All-Star before.

“He’s a really good shooter, but also a really good playmaker,” says Mike Sr., of course somewhat biased. “He grew up playing point guard. He has a shot, but the next guy has a better one and he’ll move it. He sees the floor extremely well, which is a big part of why Chicago liked him. He’s not one of those guys to be a great individual defender, but he’s a team defender. The last few years he was in the top five in the league in taking charges and the plus/minus leader on his team. That stuff doesn’t show up on the stat sheet, but it says a lot.

“I think his best overall position is two guard,” Mike Sr. said in a phone call from his home in California. “But when he went to Golden State (as a rookie) they were playing him at power forward. He’s 6-9 with two guard skills.”

So the struggles to fit into the NBA game began with a Warriors team working on its fifth coach in the last four seasons in Dunleavy’s rookie year. Eric Musselman lasted two seasons, then Mike Montgomery from Stanford two more and then Don Nelson, who typically demanded a new team and Dunleavy was traded to the Pacers in January 2007 in an eight-player deal.

The Pacers were then struggling from Ron Artest’s destruction of the team and let go Rick Carlisle for Jim O’Brien and failed to win more than 36 games in Dunleavy’s first three seasons as he continued his personal streak of never being in the playoffs.

After that breakout season in 2007-08, Dunleavy suffered a knee injury and played just 18 games the next season. He finally got into the playoffs in 2011 in the Pacers’ opening round loss to the Bulls in Dunleavy’s last season there. He then signed a two-year deal with the Bucks, where he became one of the league’s better three-point shooters off the bench and was one of the Bucks’ top players in this season’s playoff loss to Miami, averaging 12.3 points in 22.8 minutes per game and shooting almost 44 percent on threes.

“I think I do some things well that combat what Miami does, especially on the defensive end and hopefully I can have an impact when we see them,” said Dunleavy.

It’s another reason Dunleavy wanted to be with the Bulls, and you want guys who not only want you but who also want you because they see that as their best chance to win a championship. It’s no coincidence Miami has succeeded bringing in shooters every season like Shane Battier, Mike Miller and Ray Allen. Dunleavy given his age and size is every bit their equals and more.

And he answers a need that’s become a regular nagging question for the Bulls: What to do about the bench.

First it was the irrepressible and believed irreplaceable Bench Mob of 2011. Then the Bulls brought in Marco Belinelli, Nate Robinson and Nazr Mohammed and the reserve unit became even more effective. Now with Dunleavy and still hoping to add a veteran big man, the Bulls second unit looks as strong as any in the league with Kirk Hinrich, Dunleavy, Taj Gibson, Mohammed, perhaps another big man and pending the development of Marquis Teague.

Wednesday the Bulls also announced the signing of draft picks Tony Snell and Erik Murphy, who will have to earn minutes while also buying out the final season of Richard Hamilton’s contract.

But with Dunleavy, the Bulls again appear to have a versatile second unit that should be strong in team defense while also providing a perimeter shooting element like they had with Kyle Korver.

“I’ve been on some very good benches (nickname yet?) that have had impacts on the game,” said Dunleavy. “This day and age in the NBA it’s important to have a strong bench and that’s something I see us here having.”

Dunleavy is regarded, like Korver more than Marco Belinelli, adept coming off screens and spotting up. Belinelli went to the Spurs and will be a good addition for them as a good pick and roll player. But the Bulls believed Dunleavy provides more consistent play as Belinelli was often wildly inconsistent depending on matchups.

So they pursued Dunleavy from Day 1 of free agency. Dunleavy said the Bulls’ interest, obviously, was also a factor and flattering given the team’s success. He joked he missed general manager Gar Forman’s first call midnight July 1 as he was changing his son’s diaper. But they quickly huddled amidst a much sweeter atmosphere.

Dunleavy is more serious minded, but he did joke that in signing with the Bulls after playing for the Pacers and Bucks he figures he needs to close out his career with Detroit and Cleveland to have the Central Division super high five.

It’s also a Duke reunion as Dunleavy joins former Duke teammate Carlos Boozer, and Luol Deng. Dunleavy played on the 2001 Duke title team with Boozer, Chris Duhon, Jay Williams, who was selected one pick ahead of Dunleavy in the 2002 draft, and Battier.

Dunleavy was a star in that title game with three big late three pointers and led the team in scoring. But the winning stopped after college, and Dunleavy wants that stopped as well.

“I’ve been in the league 11 years and I’ve been through a lot of mediocrity,” Dunleavy said. “To be a part of this is special. I don’t take it for granted.

“We talked about their overall philosophy and then we went into me specifically, the things I can bring as far as shooting, ball movement, ball handling, playmaking on the offensive end,” said Dunleavy. “As a team defender, I’m going to always be in the right spot and be there to help guys. Just for peace of mind and happiness, the years and money became less relevant and the situation and culture was most important. Weeding through the different suitors, I definitely wanted to be on a team that has a chance to play into May and June. I think this team, barring injuries and things that happen, I think we have a chance to do that.”

“It was a big factor, obviously,” Dunleavy said of playing with Rose, with whom he shares agents. “With him over the last two years when healthy they’ve had the best record in the league. It’s kind of like the sky’s the limit. I think it sounds like he’s on pace to get back and be ready to go and you add everybody else into the mix and I think we’ve got a good, deep team and we have high expectations for this season.

“I think as I’ve gotten older I’ve become a better shooter,” Dunleavy said. “That’s just with practice, knowing how to get your shots a little better, taking good shots. Hopefully that trend continues and I think with this team, with the way guys pass and move the ball I’ll get even more looks. I don’t foresee a big adjustment because I know the way they play. I watched a lot of their games, played against them a lot. It’s a lot of the basic fundamentals. They just do it better than everybody else.

“I can get my shot off, come off screens, move without the ball, which is one of my strengths,” said Dunleavy. “In this day and age in the league you’ve got to have guys who can knock down shots but also on a close out get by people and make plays and I feel I can do that. They were looking for wing players to stretch the floor and with the culture they have, the tradition, the coaching staff, the players, it seemed ideal to come and be a part of. You picture yourself in other situations and this seemed to be the best. I’ve been looking for a situation like this for 10 years. Here it is, and I’m ready.”

Maybe not the next Bird, but the guy whom the Bulls hope can help feather their championship nest.