Iron Man once offered Wolverine a spot on the Avengers, but Captain America fought back against his inclusion for one particular reason: he's a murderer. In New Avengers #6, the superteam comes off a deadly mission, where Steve Rogers is frustrated by Logan and S.H.I.E.L.D.'s more dangerous tactics in battle. When Tony Stark suggests Wolverine is the Avengers missing ingredients, he gets immediate push back from Captain America as he sees him as a danger.
Whether he's been a member of the Avengers, X-Men, or any other Marvel superteam, Wolverine has always played by his own rules. He's not afraid to get violent when the moment calls for it - or even when it doesn't. The mutants mindset particularly irked Captain America, who has a much much clean-cut mindset about violence. Steve Rogers doesn't kill (unless it's the only option available) due to his moral code, one that came into conflict with his newest teammate during an early run on New Avengers.
In New Avengers #6 by Brian Michael Bendis, David Finch, Danny Miki, Frank D'Armata, Richard Starkings, and Comicraft's Albert Deschesne, tensions are riding high after the Avengers witnessed S.H.I.E.L.D. wipe out a part of the Savage Land and kill a number of innocent people in retaliation to what they believed was an act of war. In the ensuing conversation, Iron Man asks Wolverine if he'd like to join them as part of the New Avengers. Captain America immediately opposes the offer.
So, how does Iron Man change Captain America's mind after the hero calls Wolverine a "murderer?" Tony provides much-needed context. With the Scarlet Witch previously losing control and witnessing S.H.I.E.L.D. kill innocent people, Iron Man says they need a "samurai warrior" who is going to "go to that place we can't," suggesting Captain America knows "exactly what I mean." Reluctantly, Captain America agrees, wishing they didn't need someone like Wolverine on their team. The move proved to be a success, as Wolverine joining the Avengers changed the team forever. He brought a much-needed edge and unpredictability to the team.
On paper, Wolverine isn't the best candidate to join the Avengers. He's selfish, excessively violent, and has larger loyalty with the X-Men. However, he became the perfect candidate to join the team when the Avengers witnessed death and destruction on a level they weren't sure they could entirely fight back against. So, despite Captain America's initial objections, adding Wolverine to the New Avengers turned out to be the right move, even if there wasn't agreement in the first place in allowing him to join.