As revealed by Goal, the two United midfielders were booked in for medicals at Fulham’s Motspur Park headquarters on Friday.
Fulham are still on course to sign Tunnicliffe but the Cole deal is believed to be on the verge of collapse.
The Londoners had come to an agreement with United on Thursday to pay fees for Tunnicliffe and Cole, who are deemed surplus to requirements at Old Trafford, and capture the pair on permanent deals.
It has been a frenzied spell of transfer window activity at Craven Cottage, with Rene Meulensteen’s side having seen off interest from West Ham to seal an £11m deal for Olympiakos striker Kostas Mitroglou.
Heitinga will join after Fulham completed the loan signing of Lewis Holtby from Tottenham earlier on Friday and signed midfielder William Kvist on loan from Stuttgart on Thursday.