Twelvetrees, 23, is out of contract in the summer and, with Gloucester reportedly tracking his situation, seems to see his future away from Welford Road, much to the disappointment of director of rugby Richard Cockerill.
"Billy has played for us in every game he has been available apart from two this season," Cockerill told the Leicester Mercury.
"We have put three years of development into Billy and he was coming on nicely in our environment. If he does not want to stay and fight for his spot, that is disappointing."
Leicester are currently fourth in the Aviva Premiership, three points ahead of fifth-placed London Irish.