Albrighton revealed earlier this week that he was unsure whether he would be offered a new deal when his current contract expires in the summer, but Lambert has suggested that chief executive Paul Faulkner has begun talks with the winger's agent.
"Regarding Marc, that will get taken care of, I'm pretty sure pretty quickly," the Villa manager told reporters.
"I think Paul has spoken to his agent a couple of weeks back. I think Marc's relaxed. He'll be fine. It will get done."
Albrighton spent time on loan at Wigan Athletic earlier this season before forcing his way into Lambert's squad in recent months.