The 25-year-old has been deemed surplus to requirements at the Emirates for some time and almost completed a £5m move to Crystal Palace in January, only for the deal to fall through at the last moment.
Gunners manager Arsene Wenger is keen to offload a number of players this summer in order to trim down his squad and free up funds for new signings, with the likes of Jenkinson, Calum Chambers, Mathieu Debuchy, Jack Wilshere and Kieran Gibbs all believed to be in the frame for the chop.
According to The Mirror, Jenkinson is one of Swans boss Paul Clement's top targets and could even move to Wales on a loan deal if terms on a permanent transfer cannot be reached.
The two-time West Ham United loanee has also reportedly been on the radar of Premier League newcomers Newcastle United and Brighton & Hove Albion, as well as Serie A sides Cagliari, Sassuolo and Hellas Verona this summer.
Jenkinson joined Arsenal from Charlton Athletic in 2011 and has since made just 38 league appearances for Wenger's side.