"Stewart Downing is not for sale, it is as simple as that," Southgate told the club's official website.
"We get calls from lots of clubs about people, but we have not had any about Stewart.
"It surprises me when there is all the speculation because if people are interested in buying, it would be nice if they made contact with us and then we could just tell them straight to their face that he is not for sale.
"But that's part of the summer transfer merry-go-round. You have to deal with that speculation.
"I always say to the players, as long as they focus on fact and what they know and what I tell them - because I am very honest with them - then there is no problem."