Connelly was formally introduced on Thursday by team president Josh Kroenke.
"The last guy we talked to before coming in for this press conference was Andre," Connelly told reporters. "He's such a pro that he's out there (on the court) working out."
The Nuggets' top defender, Iguodala recently opted out of the final year of his contract after averaging 13 points a game last season.
His situation is just one of several Connelly will have to deal with as he tries to help the Nuggets become a genuine playoff contender.
The 36-year-old Connelly comes to the Nuggets from New Orleans, where he served as assistant general manager under Dell Demps. He replaces NBA Executive of the Year Masai Ujiri, who took the GM job with the Toronto Raptors last week.
Considered a rising front-office talent, Connelly inherits a team that won a franchise-most 57 games during the regular season only to bow out in the first round of the playoffs again, this time to Golden State.
Besides the potential loss of Igoudala, the Nuggets will be without Danilo Gallinari for the start of next season as the Italian forward recovers from an ACL injury.
And Connelly, with Kroenke, also will have to find a replacement for NBA Coach of the Year George Karl, who was let go on June 6 after the early payoff exit.
The Nuggets interviewed former Memphis Grizzlies coach Lionel Hollins on Wednesday and have also interviewed Indiana Pacers assistant coach Brian Shaw, but have not set a specific deadline for hiring a coach.
"I don't think we want to put a concrete timeline on anything," Connelly said. "There's the potential to talk to additional candidates, and we've talked to two great candidates. The two guys we've interviewed are tremendous, and I look forward to going into Josh's office and seeing where we are in that process."
Finding the right personality fit for the coaching job is most important to Connelly, who said he won't dictate a style or philosophy with the new coach.
"I would never try to impose any strategic restraints on a coach," Connelly said. "I'm looking for a good guy who's excited to have the job and build off where we are presently."
Kroenke, who offered Connelly the job during a Saturday night dinner, was bullish about his new hire.
"I'm really excited to have Tim here, and it's going to be a lot of fun moving forward," Kroenke said. "I hope Nuggets fans are as excited as I am. I've known Tim the last several years, and we've stayed in loose contact.
"It became a very easy decision to hire him and bring him here after a couple dominoes started to fall."
One domino was assistant GM Pete D'Alessandro leaving to become general manager for the Sacramento Kings late last week.
"Pete and I discussed the position, and I think he's going to have a wonderful opportunity in Sacramento," Kroenke said. "Once that happened, I didn't waste any time and it was a quick decision for me. There was one person I wanted to bring in and immediately discuss the job with, and that was Tim."
Kroenke said the prospect of re-signing Igoudala, the team's highest-paid player at more than $14 million, is very good.
"I think 100 percent," Kroenke said. "I had a good conversation with his agent last week. Andre is somebody we definitely want to bring back, and Andre is aware of our intentions to bring him back as well."
Connelly thanked Ujiri as he said how happy he was about the new job.
"I feel great about where the team is," Connelly said. "There's a winning culture ingrained here, and it's a heck of a team I'm inheriting. I look at the roster and think I'm the luckiest guy in the world."