According to the Daily Mirror, United wrote a letter to the Toffees to say sorry for the way they approached Moyes to take over from Sir Alex Ferguson this summer.
The paper says though that despite the apology, the relationship between the two clubs remains frosty.
“Goodison chiefs are still furious at the way they were treated by the champions over the pursuit of Sir Alex Ferguson’s successor,” explains the Mirror.
“Everton have also been stung by United revealing the Scot would replace Ferguson almost three weeks before the end of last season, ignoring the effect such a revelation would have on the end of the Toffees’ campaign.
Left-back Baines is believed to be a key target for United and they are said to have bid £12m for him earlier in the summer.
However, the fraught relationship between the two clubs means United are unlikely to land the England international unless they make a silly bid.
On the record though, United have described the way in which they replaced Ferguson after 26 years as “an exemplary job.”
“The man widely accredited as having the best judgement in football (Ferguson had a long period of time to think about the inheritance he was passing on,” said the club’s group managing director Richard Arnold
“He wanted to make sure the squad was in the right shape and the timing was done from a position of strength. That is exactly where the club is.
“When you look at the preparation that went into that and the position Sir Alex put himself in to make that transition, as an example of how to prepare, even down to how we managed the communication around it, it was an exemplary job.”