Norwich chairman Alan Bowkett told supporters on Thursday that Lambert was suing for breach of contract and was seeking between £1.5m and £2m.
But the Scot has revealed he is also the target of a counter-claim.
"I'll let him (Bowkett) say what he has got to say and we'll see what happens," Lambert said. "It will go before the Premier League managers' tribunal."
The Scot's exit from Carrow Road was a messy one, with the Canaries rejecting his resignation and struggling to agree a compensation package with Villa.
"The LMA (League Managers Association) are acting on my behalf. They supported me at the time and still are.
"What Norwich haven't made public is that they are suing me. It is disrespectful and wrong to that tribunal for me to comment any more.
"That's the way I'm going to leave it. My main concern is Aston Villa."
He added: "Am I concerned Norwich fans are only hearing one side of the story? In a word, 'yes'."