Roberto Martinez worked with Cleverley when he spent a year on-loan at Wigan in 2011 and it is believed the Spaniard sees him as a long-term replacement for 33-year-old Leon Osman.
The paper also surmises that any move could depend on whether Gareth Barry decides to sign a permanent contract at Goodison, though given they would play in different positions i'm not entirely sure why.
One player who won't be involved in any deal though is James McCarthy. After a rather bizarre story linking him with Man United was published - and later deleted - from the Sky Sports website earlier this week, Everton have since sought to play down any reports linking him with an exit.
Any deal for Cleverley would almost certainly be straight cash and given he is about to enter the final year of his contract the fee would not be too significant.
Personally I have a feeling Cleverley could be a shrewd signing by Martinez. He was unfairly pilloried by United and England fans alike for his poor form last season, but he clearly has ability.
Darron Gibson found himself in a similar situation before his move to Everton (he was once hounded off Twitter because of all the abuse he got from United supporters) and he turned out to be an inspired signing.
The United fans had seen a young Gibson score a couple of long range goals as a youngster and suddenly expected that every week, even though he wasn't that type of player.
When given a specific brief and role that played to his strengths he excelled - I would expect Cleverley to be given a similarly suitable job to fulfill at Everton.
Martinez has a lot of credit in the bank with the supporters and so any of his signings would be given a fair chance to prove themselves.
Maybe a fresh start with a manager who believes in him and supporters ready to back rather than barrack him may be just what Cleverley needs.
The desire to prove people wrong and reward those who subsequently believed in him cannot be underestimated either.