Gleeson, 25, joined the club from Salford in the autumn, but has suffered a knock in training.
Head Coach Craig Sandercock said: “He’s making slow and steady progress and we’re hopeful he’ll be ready for round one.
“He’d been training the house down so it’s disappointing.”
Rovers get their Super League campaign underway against Catalan on Saturday 3rd February.
Ireland international Gleeson has previously played for Wigan, Widnes and Wakefield.