Leeds made an official approach for Clarke yesterday, but it was quickly rebuffed by Rovers - who are determined to keep hold of the 38-year-old.
Leeds’ initial approach for Clarke was said to have fallen short of the conditions Rovers would demand to allow their young boss to speak to another club.
The Gas were quick to respond to Leeds’ interest in Clarke by offering him a new three-year contract, but they could be powerless to resist if the Championship side do increase their offer.
If Leeds were willing to pay around £200,000, Rovers would be obliged to allow Clarke to talk to Leeds about taking over as manager.
Clarke has guided Bristol Rovers to back-to-back promotions in the last two seasons, with his other managerial experience coming during his time in charge of Salisbury Town.
At the moment Steve Evans remains in charge at Leeds, and is due back into to work at Thorp Arch today, but he is expected to leave Elland Road in the coming weeks.