Snodgrass has been a key player for Leeds this season, scoring seven goals in all competitions, and Pulis has been tipped to make a move when the current campaign comes to a close.
However, Leeds are currently fifth in the Championship table and if they can secure promotion to the Premier League it is unlikely the club will consider selling one of their prize assets.
The 23-year-old Scottish international is thought to be keen to remain Elland Road if they can secure a second successive promotion and Stoke's hopes of landing the midfielder appear to rely on whether or not Leeds remain in the Championship.
Pulis does seem set to bolster his midfield in the summer regardless of whether or not he can capture Snodgrass.
The Potters boss has already been linked with a swoop for Bolton's Fabrice Muamba and new recruits in the middle of the park seem to be his priority heading into the summer transfer window.