The 31-year-old is out of favor at the club and not in the plans of manager Carlo Ancelotti.
He almost joined Boro two years ago, but went to Aston Villa at the time instead.
Now Sky Sports claims talks are underway between the relevant parties to do a deal before the Friday deadline.
Bolasie is keen on the move, as he wants regular game time.
Boro boss Neil Warnock knows the player from his many spells in the Premier League, and wants to bolster the team's attacking options.