Fulham are keen to continue strengthening before the transfer window slams shut on Tuesday night and have gone in with an offer to take Grimes to Craven Cottage.
However, Swansea have rejected the proposal put on the table by the Cottagers.
Grimes has been tipped to potentially be on the move away from the Liberty Stadium before the window slams shut.
Fulham though will have to improve their offer for Grimes if they want to sign him.
Grimes clocked the full 90 minutes on Saturday as Swansea suffered a 3-1 defeat the hands of Preston North End in the Championship.
Fulham meanwhile thrashed Stoke City 3-0 to further reinforce their promotion credentials.
The Cottagers sit top of the table after five games, level on points with West Brom but ahead on goal difference.
