The offer is worth a reported £4 million for the midfielder, who began his career with Arsenal.
Losing Lansbury would be a major disappointment to Forest fans, with no prospect of seeing the club re-invest the fee in players this summer.
This is due to a Football League imposed transfer embargo, which has remained in place for the second successive transfer window.
Forest are limited to signing only loan players and free agents.
They lost their first game of the season away at Brighton and Hove Albion at the weekend, a match in which Lansbury featured.
Burnley have been pushing to sign Lansbury for weeks, reported to be close by the Lancashire Telegraph last month.
Forest fans will have to hope Lansbury decides to stay put and hold out for a Premier League offer rather than sign for another Championship side, but if the bid has been accepted the the writing may already be on the wall regarding his exit.
More to follow...