Schmeichel was a key figure in United's dominance of English football during the 1990s and the club initially struggled to replace the imposing Dane upon his departure in 1999.
A string of goalkeepers came and went before the Red Devils settled on Edwin van der Sar, who has been a consistent presence between the posts since joining from Fulham in 2005.
However, van der Sar has been tipped to retire at the end of the season after turning 40 and speculation is mounting as to who will be his successor.
Liverpool's Reina has recently been linked with a move to the Reds' arch-rivals amid rumours that the Spaniard wants to leave Anfield.
Schmeichel believes United need to bring in someone of Reina's ability to replace van der Sar and would not be surprised to see his old club bid for the 28-year-old.
"I'm pretty sure they would go for someone like Reina, I'm pretty sure that's what they are trying to do," Schmeichel told Beyond The Pitch.
"You can look at talent, you can look at young, good players, you don't want that, you want someone that can go in straight away and give performances in 80 per cent of the games that the number one choice goalkeeper will play for Manchester United.
"Too many goalkeepers have been through in and out of the doors at Manchester United and I think now the general perception is, for this team to play well and win trophies, that position has to be someone who is of the very, very highest quality, but also with the right experience and that's why I think someone like Reina would fit the bill.
"The rumours are very, very strong around Old Trafford with this. A lot of goalkeepers are mentioned, but Reina seems to be the one that fits in the team and if he ever was to leave Liverpool, which again there is a big, big rumour he will, Manchester United would be a great club for him to come to.
"I hope that someone like Pepe Reina will go to United when Edwin calls it a day and I hear rumours that he is going to leave Liverpool so for him to go to Man Utd would be very, very good for us."