The American confirmed last week he has put the Premier League club up for sale, leading to speculation over Lambert's future at Villa Park.
But after discussions in New York between Lambert, Lerner and chief executive Paul Faulkner, Lambert issued a rallying cry to supporters concerned by a flirtation with relegation that saw them finish 15th in the Premier League.
"As everybody is aware, I met with the chairman along with Paul and I'm determined to get on with the job of managing Aston Villa," he told the club's official website.
"These circumstances are extremely challenging as the club is for sale and as far as I know there is nothing imminent in terms of new ownership, but we're going to roll up our sleeves, prepare for the battle ahead and do our best by the club.
"The squad requires strengthening, that's self-evident, and it won't be easy within the current circumstances. But we'll look to identify players of proven quality to bring in and get ready for the start of the new season.
"To the fans, my message is simple - this is your club and it's a fantastic club. Your support has made it into this. The past two seasons have been tough and expectations, rightly, are high.
"But success is always built on everybody working together and the circumstances now require us all to pull together and stick together and try to galvanise ourselves for this challenge we face.
"I'm determined to give it everything I have, the players will do the same and it is our collective strength as a club which, ultimately, will see us through."