The 27-year-old Australian completed his free transfer from Aston Villa to become the second player, after Steve Staunton, to move between Premiership clubs under the Bosman ruling.
He said: "The chance of replacing Peter was the main bait for me to come to Manchester United - not something I was frightened of."
In signing for United Bosnich has turned down chances to play in Italy and Spain, where he was offered more lucrative contracts, but the deal he has signed is believed to be worth almost pounds 7m over four years.
United's chairman, Martin Edwards, described Bosnich as "a very brave man" to take over the mantle of Schmeichel, but the player who left Old Trafford seven years ago because United could not renew his work permit said: "I could not have found myself sitting in Italy or Spain, having turned down the chance to follow Peter Schmeichel. I could not have looked at myself in the mirror if I had done that."
Bosnich will officially join the European champions on 1 July when his contract at Villa runs out.
His departure from Villa Park has caused some anger among fans of the Midlands club, but Bosnich said: "If there is any animosity it doesn't come from my end. I would just like to thank the Villa fans for the way they've treated me while I have been there. From my point of view I have only good memories of Villa. They gave me the chance to fulfil my dreams of playing in the top flight in England."
After leaving United in 1992, Bosnich played for Sydney Croatia. On his return from Australia he married his English girlfriend and then the need for a work permit disappeared.
Edwards said: "The manager decided on Mark as the replacement for Schmeichel some while ago. But we have been unable to talk to him until the end of the season, although foreign clubs have been able to. We had to sit and wait and hope that Mark didn't lose patience and go somewhere else."
Bosnich could make his United debut on a pre-season tour to Australia, but he will step into Schmeichel's role in earnest when the club start the defence of their three trophies in August.