The Daily Express says Lambert had a second meeting in 72 hours with Lerner yesterday to try to solve the club's relegation crisis.
Lambert had met Lerner, who is in Birmingham on a rare trip from the USA, on Friday and talks that also included chief executive Paul Faulkner have continued.
Lerner has already forked out £25million on young players this season but the ambitious plan aimed at ending two successive years of fighting relegation has not worked, with Villa again in grave danger of falling out of the Premier League.
They have won only four league games all season and have been battered in their past three fixtures, where they have conceded 15 goals.
But Lambert is being told by Lerner that money is tight.