The pending sale of the 26-year-old midfielder was thought to be one of the contributing factors to Steve Cooper deciding to leave Swansea last week.
Fulham are set to complete the signing of Swansea midfielder and captain Matt Grimes
He is due to undergo a medical next week.
Fulham are continuing to build a capable squad for new boss Marco Silva having signed Harry Wilson from Liverpool in a £12m deal at the weekend and former Tottenham goalkeeper Paulo Gazzaniga.
Grimes’ impending departure played a key part in manager Steve Cooper’s decision to leave
Swansea are looking for a replacement for Cooper after John Eustace chose to stay at QPR
Swansea are looking for a new manager to succeed Cooper after leading candidate John Eustace opted to stay at QPR.
Eustace had an interview with Swansea last week and provisionally agreed to join but had second thoughts over the weekend.
So far in this transfer window Swansea have brought in Liam Walsh, Kyle Joseph and Joel Piroe.
The club’s pre-season preparations have been affected by positive coronavirus cases, with the friendly against Bristol Rovers having been cancelled.
Swansea get their Championship campaign underway on August 7 when they face Blackburn away.