The Clarets were unsuccessful with a £3.5million bid for the central midfielder last summer following their promotion to the Barclays Premier League. That approach resulted in Forest tying Lansbury to a new three-year contract.
Burnley returned during the January transfer window with an improved offer of £4million and that was also turned down.
It is understood that the Lancashire club, now back in the Sky Bet Championship after just one campaign in the top flight, have this time only offered around £2million for the 24-year-old.
Needless to say, the latest bid falls some way short of the value placed on the player by Forest, who are under a transfer embargo for breaking Financial Fair Play rules and would therefore be unable to spend any proceeds from the sale of Lansbury.
Lansbury is an important player for the Reds, having been made captain last season, while scoring 11 goals from midfield.
Forest paid £1million in August 2012 to bring him to the City Ground from Arsenal and he has since made 88 Championship starts and 12 substitute appearances, scoring a total of 22 goals.
Earlier this month Forest also rejected a fresh offer from West Brom for Michail Antonio.
It is thought the offer was worth in the region of £5million, six months after the Baggies offered £4million for the 25-year-old winger during the January transfer window.
At the time, Reds owner and chairman Fawaz Al Hasawi said on Twitter: “I would like to reassure our fans that I have no intention of selling Michail Antonio or any other player who wants to be part of this club.”