Full back was open to following Moyes to Old Trafford.
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Everton defender Leighton Baines reportedly held talks with the club over the summer and told them that he wanted to join Manchester United, the Daily Star reports.
United boss David Moyes had worked with both Baines and Marouane Fellaini during his time in charge at Goodison Park, and had been chasing a double deal for the pair, but instead manage only to land the Belgian midfielder on transfer deadline day.
The Toffees stood firm on Baines, though, and in the end the defending Premier League champions were forced to fork out £27.5m for Fellaini.
However, the club coach and former defender Alan Stubbs admitted that the player had been keen on following his former boss to Old Trafford this summer.
“Bainesy was never really an option to leave,” Stubbs said.
“The club had dug its heels in but Bainesy had obviously said he’d like to go.”
United are believed to be keen on returning with an improved offer for the England international during the January transfer window and Baines is likely to still be keen on making the move.
Baines struck twice during last weekend’s 3-2 victory over West Ham United.