Blues rejected an initial bid of £300,000 from their Championship rivals earlier this month but the latest offer, which is believed to be much closer to their valuation of the winger, may be accepted.
Reds boss Alex McLeish is a long-time admirer of Burke having given the Scot his chance at Rangers before luring him to Birmingham in 2011, although he left the club just days later to take over at Aston Villa.
McLeish is now looking to link up with the 29-year-old wideman for a third time as Forest strive to strengthen their hopes of securing at least a play-off place.
Burke's potential exit is not expected to be the only one from St Andrew's before the close of the January transfer on Thursday night, with the likes of Jack Butland, Curtis Davies, Marlon King, Nikola Zigic and Nathan Redmond all heavily linked with moves away.
Financially-stricken Birmingham are in need of funds as they look to stave off the possible threat of administration, with every member of manager Lee Clark's squad available at the right price.