The move to Crystal Palace would see Downing reunited with Sam Allardyce, the manager who signed him from Liverpool for £5 million in 2013 and spent two seasons with at West Ham United.
Downing opted to drop down to the Championship last season to help his hometown club win promotion back to the Premier League.
However, the former Aston Villa and Liverpool winger has had limited chances in the starting eleven for Boro, spending much of this season on the substitute’s bench.
Aitor Karanka has opted for players such as Viktor Fischer, Adama Traoré and Christian Stuani ahead of Downing, with the 32 year old searching for regular first team football.
Speaking about the matter, the Boro boss said that "a player who wants to leave, I am always trying to buy a player, to sell a player, to bring in a player on loan or sell. If Stewart wants to leave and everybody is happy, yes of course."
"If the club thinks it’s good for the club and for the player, I can’t say anything," he said.