Graham was deemed surplus to requirements at the Stadium of Light last summer after failing to score in 13 appearances following his £5m transfer from Swansea City a year ago.
He departed on a season-long loan to Hull City where it took him 14 games to break his duck for the Tigers.
The onus is on Hull rather than Sunderland to find Graham a new club because the forward is contracted to the Humberside outfit for the rest of the season.
Bolton boss Dougie Freedman is desperately keen to bring in a player he feels will be able to get on the end of the chances his struggling side are creating.
But it is likely he will have to be patient and hope promotion-chasing Nottingham Forest do not make a more attractive offer.
Hull’s priority is to offload a player who they no longer need and whose sizeable wages they are still helping to pay.