However, he insisted there had been no developments on either front.
He said: "It's speculation, simple. There is interest always in good players, but we have enquiries throughout the year and there has been nothing firmer than that.
"There is nothing, really, that has been anything out of the ordinary, in my opinion."
Asked if he feared Downing had played his last game for the club, Southgate replied: "I don't think so, no.
"It's January and it is a month I will be glad to see the back of. There is very little to report."