Van Persie chose United over title rivals City when he left Arsenal in a reported £22m deal last summer.
Mancini has previously admitted he was interested in signing the 29-year-old but has only now indicated how strongly City pursued him.
The Italian said: "We wanted Van Persie because we knew he could be an important player.
"He is totally different from our other strikers. We wanted him for the Champions League and for the Premier League.
"We were very close. We were sure he was for us but this did not happen and now we can do nothing. We were very close three or four months before he joined United.
"I think Van Persie, for United, is an important player. It is clear he changed this situation. In this moment, the difference is this."
Mancini has often bemoaned the output of his own strikers in recent weeks.
Sergio Aguero, Carlos Tevez, Edin Dzeko and Mario Balotelli have all struggled for goals of late - a problem highlighted in the shock Boxing Day loss at Sunderland.
Dzeko responded by scoring twice at Norwich on Saturday and Aguero was also on target in a 4-3 win.
Mancini insists he retains faith in his attacking options even though he did not get the player he wanted last summer.
He said: "I am happy, very happy. Last year we were the team that scored the most goals in the Premier League.
"But when you can improve your team with another top player, this is important.
"With Van Persie we would play with three strikers sometimes."