Potters boss Alex Neil feels he needs more final third options and a number of potential arrivals are being assessed by the club.
Blackburn’s Gallagher, who the Ewood outfit could sell at the right price, is a player Neil likes.
However, a swoop for Gallagher, 27, is not imminent and it is suggested he is a player Stoke would likely look at later in the window if they still need a striker.
Blackburn are claimed to be looking for around £5m in order to let Gallagher go, a price Stoke are unlikely to want to pay.
Stoke are not alone in their admiration of Gallagher, with Coventry City also holding an interest in him.
Gallagher, who is on a lucrative contract at Ewood Park, joined Blackburn in 2019 from Southampton.
The striker made 34 appearances in the Championship for Blackburn last season and hit the back of the net on seven occasions, including against Stoke and Coventry.
