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.
Everton are interested in signing Manchester United midfielder Morgan Schneiderlin.