The French-born star emerged as a shock target for the Teessiders on Tuesday and the club's need for a winger has since increased significantly after Adama Traore's move to Wolves was confirmed.
The aformentioned source claims that Bolasie's former club Crystal Palace are also keeping tabs on the situation but that for now, Middlesbrough remains his most likely destination.
Tony Pulis admitted following Tuesday night's 3-0 thumping of Sheffield United that he has "quality" targets, with the club understood to be discussing both a permanent deal and loan with a view to a permanent move as they leave no stone unturned.
Bolasie only joined Everton from Palace in August 2016 for £25m, but his stock has dropped significantly since then and he is now not expected to fetch more than £15m.