The English Championship clubs are understood to be weighing up loan offers for the Northern Ireland international, according to the Daily Mail.
Jones hasn't played for the Light Blues since he injured himself in a red-card challenge on Celtic defender Moritz Bauer in an Old Firm clash in September.
Although the former Kilmarnock man is thought to be nearing a return to action, and has featured for the club's reserve side in recent weeks, he may be allowed out on loan next month to get gametime, with a number of players ahead of him in the pecking order at Ibrox.
Stoke's current manager is Michael O'Neill, who gave Jones his international debut while the 25-year-old began his career at the Riverside but only made one senior appearance for Boro.
Jones made a handful of appearances in the Europa League qualifying rounds for Steven Gerrard's side, featuring against St Joseph's, Progres Niederkorn, FC Midtjylland and Legia Warsaw, and also got minutes against former club Kilmarnock, Hibs and St Mirren before his substitute appearance and eventual dismissal against Celtic.
He also played against East Fife in the Betfred Cup round of 16.