Top spy Brian Talbot saw left-back Tierney, 21, in fine form for Brendan Rodgers' side in European action last week.
Now the Craven Cottage side are waiting to see if he can leave this window.
Everton, Wolves and Bournemouth have all shown interest in Tierney but he is not pushing to go.
The Cottagers are set to splash out as they know they have a big sale up their sleeve in £100m-rated Ryan Sessegnon — if they need to raise cash later.
Ideally they would like both in their side for the return to the Prem but landing Tierney would soften the blow if Sessegnon moves in the future.