The Manchester United midfielder joined the Villans on a season-long loan last Tuesday, one day after the transfer shut but it emerged that both clubs had already filed the paperwork.
Cleverley is thought to have initially snubbed a deadline-day move to Villa Park upon hearing that the Toffees were interested in him, but Lambert insists that was not the case.
"My main concern was that he wanted to come here, he wanted to play," he told reporters.
"I know all this speculation about him not wanting to come, that's not the impression when I spoke to him.
"The deadline day is mayhem, things can go wrong, be on and off, carnage breaking out everywhere. I knew the situation. The lad never indicated he didn't want to come. I spoke to him, he was professional about it."
The 25-year-old could make his debut in Saturday's trip to Liverpool.
