The 32-year-old is thought to be looking elsewhere for the possibility of first-team football having started just one Premier League game for Boro since November 5.
"I have said always that a player who wants to leave and Middlesbrough Football Club and another club agree the transfer..." he told reporters.
"To find or to sell a player, to buy a player, to sign a player on loan, to bring in a player on loan from other places, all parties have to agree.
"If Stewart wants to leave and the clubs agree and everybody's happy, then yes, of course. If the club thinks that it's good for the club, good for the player and good for the club, I can't say anything."
Downing, who is currently in his second spell at the Riverside, has scored one goal in 17 league games for the club this season.