Jose Mourinho was keen on signing the England international last summer but United refused to meet Leicesters £75million valuation of the centre-back.
Manchester City boss Pep Guardiola is an admirer of Maguire and believes the former Hull City man could fill the void left by Vincent Kompany, who announced his departure from the club earlier this month.
However, Sheringham believes that the defender is perfect for Old Trafford.
Well, I think Real Madrid are going to be in the same boat as Manchester United this summer, so youve got to beat them to sign the top players that are out there, said Sheringham.
Central defence is a big area where they need to find a Vincent Kompany or Tony Adams-type player
I really believe they should go back to the British mentality. The core of any good, British team, comes from British players, as I see it. Over the last 20 or 30 years, if a British team does well in the league or in Europe, theres always a core of British players involved.
OK, Manchester City do go away from that trend, but in general they are players with a British mentality who understand what it takes to win. You only have to look at the past success of Manchester United, Arsenal and Chelsea , to see that core of British players.