The Lancashire side are said to be determined to sign the midfielder regardless of if they remain in the Premier League or are relegated to the Championship.
The report states: "Dyche is poised to offer around £5million for Lansbury whether the Clarets stay up or not as he is desperate to land the former Arsenal prospect."
The 24-year-old has been an important part of the Nottingham Forest side this season, making a valuable contribution.
Lansbury is a former England under-21 international, and is Forest's third top scorer this season behind Britt Assombalonga (15) and Michail Antonio (14).
Forest are unlikely to consider letting him go lightly, but £5 million is a substantial amount for any side in the Championship.
Last summer the club cashed in on the duo of Jamaal Lascelles and Karl Darlow, selling the pair to Newcastle, but were swayed by being allowed to keep them on loan for the season.
Burnley's chances of persuading Lansbury to make the switch would be substantially aided if they can retain their Premier League status. They are bottom of the league at present but only three points away from safety as the table currently stands.