The Scottish manager's future at Villa Park is in doubt following a mediocre campaign which has the West Midlands outfit sitting 15th on the Premier League table with one match remaining.
As owner Randy Lerner gets set to make some key decisions over the summer period, Lambert has spoken of his desire to stay in charge of the Villains next season and beyond.
"I would love to be. It's a brilliant club with a fabulous fan base," he told Sky Sports.
"Any club that wins a European Cup creates history and tradition and this club has got it in abundance.
"This team has got to live up to it and at the minute we haven't done it."
Villa lost 4-0 at Manchester City on Wednesday night and although the scoreline was one-sided at the final whistle, Lambert felt his team were in the contest for the most part.
"I thought for at least an hour we were frustrating them and the game plan worked perfectly," added Lambert.
"I thought we were excellent in the first half and had a few little chances to score.
"With the injuries we've got to big players, I could not fault them."
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