The 29-year-old has attracted interest from both clubs as they look to bolster their squads ahead of the loan window closing on August 31, with the wideman in demand.
Bolasie joined Everton from Crystal Palace in 2016 but a crucial ligament rupture in December of that year saw him out of action for 370 days. He has made 29 appearances in the league for the Toffees but has managed only two goals and four assists in that time.
Villa boss Steve Bruce has been vocal in his desire for the Birmingham-based club to add another player or two to his squad before the loan market closes, with the manager stating he wants a wideman and a striker to add depth to his squad.
However, Middlesbrough look set to win the race for Bolasie signature, with the African attacker to replace Adama Traore at the Riverside Stadium, who joined Wolves earlier this month.