Walsall have made their feelings known to the Football League but City have denied Lambert had any influence in the decision.
Chief executive David McNally said: "We wish to point out the only person ultimately responsible for the decision to call off the match is the referee.
"Whether we agree with his decision or not is immaterial. The responsibility for deciding whether the fixture can go ahead is his and his alone.
"We are surprised and disappointed that (Walsall) appear to be questioning his professional integrity over this matter."