Clarke has attracted an official approach from Leeds after United’s interest in MK Dons’ manager Karl Robinson came to nothing.
Bristol Rovers met United's approach with a list of conditions, including the need for an agreement over a fee in compensation. The club are awaiting further contact from Elland Road before allowing Clarke to open talks with United.
The 38-year-old holds a 12-month rolling contract at the Memorial Stadium but Rovers offered him an improved three-year deal yesterday in an effort to prevent his departure.
A statement issued by Bristol Rovers today said: "Bristol Rovers can confirm that they have received an approach from Leeds United to speak to Darrell Clarke regarding him taking over as manager at Elland Road.
"As yet, the Yorkshire side have failed to meet the conditions set out by Rovers to enable any talks to take place and until that happens there is nothing else to report.
"We can also confirm that Rovers have offered Darrell a new and improved three-year contract to remain as Bristol Rovers manager."
Clarke is among a list of lower-league coaches singled out by Leeds as potential replacements for existing head coach Steve Evans and his chances have increased on the back of unsuccessful talks between Robinson and United owner Massimo Cellino.
Robinson, who has managed MK Dons for the past six years, spoke with Cellino in London last week but emerged from that meeting with doubts about the position at Elland Road and the prospect of working under the controversial Italian.
Sources at Elland Road told the Yorkshire Evening Post yesterday that Robinson was “very unlikely” to be installed as Evans’ successor.
Leeds have already looked at Oldham Athletic manager John Sheridan, the ex-United midfielder, but the club did not follow up initial enquiries with an official approach to speak with him.
Clarke, however, is firmly on their radar having led Bristol Rovers to back-to-back promotions from the National League and League Two.
The former Hartlepool United player has managed Rovers since 2014 and won promotion back into the Football League at the first attempt.
His squad claimed automatic promotion to League One at the start of this month with a dramatic injury-time goal on the last day of the season. He was named League Two manager-of-the-month for March after six wins from seven games.
Clarke is one of the youngest manager in the Football League and his only other coaching job was held at non-league Salisbury City, the club where he finished his playing career.
United’s search for a new head coach is continuing despite the fact that Evans officially remains in post at Elland Road.
The Scot has a deal until the end of June and is due back at work tomorrow after a short break abroad but he looks poised to part company with the club after 38 games in charge.
The YEP understands that people close to Evans are now advising him to quit United in the wake of the approaches for Clarke and Robinson.
Evans said: “I’m refreshed and l’ll be working until someone tells me different.”