Downing told to forget quitting Middlesbrough
Boro had opened talks with Tottenham over Downing's transfer earlier this month, but the deal has dragged on and the move is now dead.
"Stewart is not for sale, both myself and the chairman Steve Gibson told him that. He is a very important player for us," said Southgate.
"Stewart met with the chairman on Thursday and is probably letting it all sink in.
"January is an unsettling period for many players. Some think they are on their way when they aren't and some want to be on their way and end up staying.
"As far as we are concerned with Stewart, it is dead."