That's according to Chris Bascombe, The Telegraph's northwest football correspondent who reports that while the French midfielder could be retained by the new Everton boss, Ashley Williams, Davy Klaassen and Kevin Mirallas are not part of his plans.
Schneiderlin had a poor season overall in 2017-18 as the Blues struggled in the Premier League before eventually finishing eighth. The former Southampton and Manchester United midfielder's commitment was questioned but his form picked up towards the end of the campaign, with some observers suggesting that, having informed Everton of his desire to leave this summer, he was putting himself in the shop window.
Silva's appointment may have changed that and it could see Schneiderlin stay at Goodison Park if Bascombe's information is correct.
The 33-year-old Williams, who didn't play a game for Everton after being sent off at Burnley in early March, expensive misfit Klaassen and the enigmatic Mirallas, meanwhile, could be on the move before next season kicks off.
Like Schneiderlin, Mirallas was sent home from training during David Unsworth's caretaker-manager spell last autumn for reported indiscipline and his loan to former club Olympiakos was terminated early due to poor performances.
Klaassen, meanwhile, failed to start a game between December and the end of the season and has been the subject of speculation linking him with a loan move to Turkish side Besiktas.