The Evening Chronicle says Woodman will have one year left to run on his Toon deal this summer and is not interested in playing the role of back-up to Martin Dubravka and Karl Darlow, which is the scenario currently ahead of him when his loan deal with Swansea is over.
Arsenal have scouted Woodman regularly this season in the Championship where be as been one of the stars of the season.
The Swans would love to sign Woodman permanently but aren't guaranteed promotion.
Before football's enforced break, Steve Bruce had said: “We're very, very fortunate that we've got three or four good goalkeepers.
“The kid at Swansea – who we're monitoring – is doing very, very well."