Lens has been a peripheral figure ever since the 61-year-old boss was appointed – and he has not made a Premier League start since October.
The 28-year-old was left out of the Black Cats’ squad for their most recent match against Chelsea for disciplinary reasons, and he admits that he could make a swift exit from Wearside in the January transfer window.
"It's definitely annoying,” he said. “I am not playing and I did not come to the Premier League to sit on the bench. Something will have to change anything. What, we will see though.
"A transfer in the winter? That could happen. Once the window is open, we will see what will happen."
When Lens was brought to Sunderland there was a great excitement over his arrival, and his initial performances were impressive.
The former Dynamo Kiev man showed good pace and energy in his initial displays for the Black Cats, but he has regressed quickly.
Allardyce is said to have become concerned by his attitude, and the Daily Mail suggested that Lens had aggravated his teammates with his lack of effort in training.
Lens has made 13 appearances for Sunderland so far, but he has only hit the back of the net on two occasions.