Ronaldo wanted to leave Old Trafford in the summer after United failed to qualify for the Champions League this season.
The Portugal icon returned to Manchester United in the summer of 2021 and mostly impressed last season, scoring 24 goals in all competitions.
However, new United manager Erik ten Hag appears to have misgivings about Ronaldo, having only started him in two of United’s nine Premier League games this season.
While Sheringham believes the situation is less tense than it was early in the season, he believes Manchester United should have sold Ronaldo during the previous transfer window.
Ronaldo’s departure would have left a gaping vacuum in United’s locker room, but Sheringham believes Tottenham striker Harry Kane would be the ideal replacement at Old Trafford.
Kane, who tried to leave Tottenham the same summer Ronaldo returned to England, has 274 goals in 465 appearances for Tottenham and is closing up on Wayne Rooney’s all-time England record.
‘If they’re going to go out and break the bank for someone, go out and get Harry Kane,’ Sheringham told Gambling.com.
‘He’s the type of centre-forward they lack, a proper professional, he leads the line and you’ll always get 100% out of him.
‘He’s a proper player, so go break the bank for him. I know that’s going to upset the Tottenham fans but we’re talking about Manchester United, one of the biggest clubs in the world.
‘They should be doing this sort of thing instead of spending money on players who only might do the job for you.’
Asked whether he believes Kane could be tempted to leave Tottenham, Sheringham added: ‘Harry looks very happy to me.
‘He obviously had a wobble at the start of last season when he thought he might be leaving the club, and you could see he was unnerved by all that.
‘He’s in good shape at the moment, has the World Cup coming up, he’s scoring goals, so he’s in fine fettle so I don’t see why he would be thinking about leaving.’
Ronaldo has scored just one Premier League goal this season and was subbed off during Manchester United’s goalless draw with Newcastle United last weekend.