The France international turned 30 last month and it remains to be seen how sustained the Gunners’ interest will be.
Sources have told Football Insider that Shelvey has come to the attention of the club’s scouting team as he wins plaudits for his passing range and ability to dictate the tempo of top-level matches.
Moreover, Jack Wilshere will be sidelined for another three months as he prepares to undergo surgery on his fractured left fibula.
Arsenal were impressed with the way Shelvey performed in the closing months of last season, including when Garry Monk’s team won 1-0 at Emirates Stadium in May.
The 23-year-old has begun the new campaign in fine form, and impressed on his return to the England team, in both the victories over San Marino and Switzerland.
It was his first appearances for the national side in three years and demonstrated the strides that Shelvey has made since he was publicly criticised by Monk last December for being “lazy” and ill-disciplined.
The player subsequently hired a personal trainer and his own in-house chef in a bid to improve his fitness and lifestyle.
The former Charlton and Liverpool midfielder, who joined Swansea from Anfield in 2013 for a cut-price £4million, is once again attracting attention from the major Premier League clubs.
Football Insider reveals the latest Arsenal transfer news after manager Arsene Wenger was widely criticised for his club’s summer transfer activity, with the Gunners the only team in Europe’s five major leagues not to sign a single outfield player.