Downing, who is currently sidelined through injury, is one of a number of Boro players expected to move on in the wake of their relegation from the Premier League.
The England international was linked with a move to Tottenham in the January transfer window with Downing having a transfer request rejected by the Teessiders.
Gibson will not stand in Downing's way this summer if they receive a suitable offer, but appears to have ruled out a switch to Tottenham for the winger insisting a deal will only happen if it one of the top four that comes in for the player.
"We had an unspoken agreement that if the right club came in, and I believe Stewart is a top-four club player, then we wouldn't stand in his way. It's a short career," said Gibson.
"There comes a point where a player must sit down and, without emotion, make a decision about their career.
"He has five or six years left at the top. Can I offer him European football? No. Does he deserve European football? In my opinion, yes."