Lambert is said to be upset at the owners' plans to cut costs next season, rather than backing an assault on promotion.
The former Aston Villa boss, who led Norwich out of the second tier five years ago, replaced Gary Bowyer at Ewood Park in November.
However, Rovers have slipped two places to 18th in that time, with Lambert's plans hit by the protracted sale of top scorer Jordan Rhodes in February.
Blackburn have made a number of short-term signings in the meantime, but further reinforcements will be required if they are to compete at the other end of the division next term.
It is set to be a summer of change at Ewood Park regardless, with the loans of Jordi Gomez, Matt Grimes, Danny Graham and Tony Watt coming to an end, although Rovers have an option to turn Watt's deal into a permanent one.
And the Lancashire side have six first team players - Matthew Kilgallon, Tommy Spurr, Lee Williamson, Simon Eastwood, Simeon Jackson and Chris Brown - out of contract.