Oxlade-Chamberlain has entered the final year of his contract at Arsenal, where he could soon be joined by Monaco and France star Thomas Lemar, according to the tabloid.
Arsenal's Alex Oxlade-Chamberlain
It would appear, therefore, that the England international will be on his way out of The Emirates this summer, and Liverpool and Chelsea are both said to be keen.
But The Sun reports that Stoke have also entered the running, after losing Marko Arnautovic in a £24.5 million deal last week.
Though they must surely be considered outsiders at this stage, any Stoke interest could appeal to Oxlade-Chamberlain, whose Burslem-born father, Mark Chamberlain, played for the Potters for three years in the 1980s.
Chamberlain later moved south with Portsmouth via a spell at Sheffield Wednesday, and raised his son in the Hampshire city.
Former England footballers John Barnes (left) and Mark Chamberlain
Despite his father's Portsmouth association, Oxlade-Chamberlain began his career at Pompey's fierce rivals Southampton, before being signed by Arsenal in a £12 million deal six years ago.
Could Stoke tempt Arsenal's Oxlade-Chamberlain?