The Daily Mail says Tottenham have been linked with the player but what Downing is told next week could hold the key to his immediate future.
Manager Gareth Southgate said: "We are very pleased with Stewart's progress. He underwent a serious foot operation, but he is determined to come back stronger and fitter as soon as he can.
"He is due to see a specialist in London on July 7 and we expect him to be given the go-ahead to step up his rehabilitation from there.
"So far, it's been as much about resting and allowing the foot to recover from the operation, but I know he will be keen to focus on some more serious rehab and make real strides towards getting fit."