Owen Coyle is a long-term admirer of Russell and the midfielder's agent, Gary Mellor, claims the Clarets boss has already failed with one bid for the 28-year-old.
Russell, who has one year to run on his current contract, was left at home for Norwich's tour of Scotland as the club prepare for life without him and Mellor says there is no shortage of clubs waiting to take a punt on the former Stoke man.
"I understand Burnley made a bid for Darel that was turned down by Norwich," Mellor told Sky Sports.
"Burnley tried to sign Darel last year and I know Owen Coyle is a big fan of him.
"Burnley are not the only club chasing Darel as there are several Championship sides keen on him, including Preston.
"Darel has one year to run on his contract and he has enjoyed his time at Norwich, but he would like the chance to play at the highest level.
"It is unlikely Norwich will keep Darel and we hope a solution can be agreed between all parties."