"We have to accept there will be speculation about our players and, if they're the quality of Stewart and have had the sort of season he has, that's inevitable," Southgate has said.
"Stewart knows exactly what I want to try and achieve here and, as far as we're concerned, he's not for sale.
"I fought long and hard to keep him and get him on a new contract during the course of last season.
"I think we saw an increased maturity in his performances towards the end."
Downing recently committed his future to Boro by signing a new five year contract at the club after reports that Tottenham were interested in signing him.