The Daily Mail reports that Everton manager Ronald Koeman is keen to push a deal through for the France international, with the Toffees understood to have already tabled a bid for the player below the £20m-mark.
But United have made it clear they will not consider selling the player unless they receive a bid closer to the £24m they paid to sign him from Southampton in 2015.
Schneiderlin has only played 11 minutes for United in the Premier League this season and was the subject of a £15m bid from West Brom last week.
Red Devils manager Jose Mourinho has said he is willing to let the Frenchman leave Old Trafford – along with Dutch winger Memphis Depay – if acceptable offers for the duo come along.
"I will allow both of them to leave – if the right offer comes. Until this moment, no," Mourinho was quoted as saying by the BBC.
"We don't have any offer that is close to the quality of the players we have."
Everton owner Farhad Moshiri is reportedly keen to back Koeman in the transfer market and is hoping to complete deals for Schneiderlin and Depay from United and Virgil van Dijk from Southampton in January.
Koeman asked the Goodison Park hierarchy to make new signings in the wake of Everton's 2-1 defeat to Leicester City in the FA Cup.
"I know what is needed to change," he was reported as saying by the BBC.
"If everybody opened his eyes maybe we will get further in our improvement – because that is what this team needs.
"I have that in mind and the board knows because we speak together about what we and the team needs and that is the job of [director of football] Steve Walsh.
"I hope everybody realises what the team needs to get results that are the expectation of this football club."