Contact has been made between the two clubs, although there is no agreement over a deal yet.
The France player, who signed for United for £25m in July 2015, would be reunited with former Southampton manager Ronald Koeman at Goodison Park.
Red Devils boss Jose Mourinho will sanction any deal for the 27-year-old, who is not part of his long-term plans.
Schneiderlin has played 11 minutes of Premier League football this season and he has not been included in the United squad for Wednesday night's league match at Crystal Palace.
Koeman was keen to keep the Frenchman at Southampton during his time as manager in 2014, and the Dutchman views him as a key component in his plans to strengthen Everton's midfield.
It is also understood Koeman is keen on United and Netherlands forward Memphis Depay.
The Dutchman has also attracted interest from Germany and Italy, with Roma among the clubs keen on the 22-year-old, who joined United from PSV Eindhoven for £31m last year.

He was the Spectator magazine's editor, when it carried an article which said "drunken fans" had "mindlessly tried to fight their way into the ground".
Apologising, he said he was glad a report "lays to rest the false allegation" about the fans' behaviour
The Hillsborough Families Support Group said it was "too little, too late".
Mr Johnson said: "I was very, very sorry in 2004 that the Spectator did carry an editorial that partially repeated those allegations, I apologised then and I apologise now.
"I do hope the families of the 96 victims will take some comfort from this report and that they can reach some sort of closure."
The Spectator article, which was not signed, was written in the wake of the death of British hostage Ken Bigley in Iraq, but went on to criticise "drunken" Liverpool fans at Hillsborough.
Mr Johnson said sorry at the time and went to Liverpool to reiterate his apology.
But Margaret Aspinall, chairwoman of the Hillsborough Families Support Group, who lost her son James, 18, in the tragedy, said: "What he has got to understand is that we were speaking the truth for 23 years and apologies have only started to come today from them because of yesterday.
"It's too little, too late.
"It's fine to apologise afterwards. They just don't want their names in any more sleaze. No, his apology doesn't mean a thing to me."
Everton are interested in signing Manchester United midfielder Morgan Schneiderlin.
[NEXT_CONCEPT]
London Mayor Boris Johnson has said he was "very, very sorry" for comments made in 2004 about Liverpool fans in the Hillsborough disaster.