Graham has found first team appearances hard to come by at Sunderland, and he looks to be way down in the pecking order.
The likes of Jermain Defoe, Steven Fletcher and Duncan Watmore are all ahead of Graham in the competition for places at the Stadium of Light, and Sunderland are said to be willing to allow the 30-year-old to leave.
And Blackburn are prepared to make an offer to the Premier League side, as they feel Graham could enhance their attacking threat.
Rovers have made improvements under Paul Lambert, but the former Aston Villa boss is keen to bring in more quality.
Graham has previously proven his class in the Championship with Watford, but his form in recent years has not been particularly encouraging.
Graham still only has one Premier League goal to his name, since signing for Sunderland back in January 2013.
If the experienced front man does move to Ewood Park, he will be competing with the likes of Jordan Rhodes and Bengali Koita for a place up front.