Lambert, who took charge at Norwich in 2009 and led them to promotion from League One and the Championship in successive seasons, will be announced as Villa's new manager on Wednesday.
Norwich finished 12th under Lambert in their first season back in the top flight, and had been linked with another Scotsman, Cardiff manager Malky Mackay, who made over 200 appearances for them from 1998-2004.
However, the Welsh club insisted no approach had been made and
Mackay ruled himself out,
declaring his intention to remain with the Bluebirds.
Hughton, the former Tottenham and Republic of Ireland defender, was assistant to Martin Jol at White Hart Lane before moving to Newcastle, initially as coach, then caretaker manager.
He replaced Alex McLeish as
Birmingham boss in June 2011
and took them to fourth place in the Championship, a point above Blackpool, but Ian Holloway's side beat them 3-2 on aggregate in the two-legged play-off semi-final.
Recent Norwich managers
Aug 2009 - June 2012
Oct 2007 - Jan 2009
Oct 2006 - Oct 2007
Dec 2000 - Oct 2006
A statement from Birmingham confirmed: "Norwich City have made an approach for Chris Hughton with regards to their vacant managerial position.
"Hughton expressed a wish to speak to the Canaries and permission was reluctantly granted."
Acting Birmingham chairman Peter Pannu said: "Chris is a great manager and deserves the opportunity to speak to Norwich, which is why we have granted him permission to speak to them."
Former Norwich player and coach Doug Livermore, who was a coach at Tottenham during Hughton's time at the club, told
BBC Radio Norfolk:
"I think Chris Hughton would be a very, very good appointment.
"When you look at the budget Newcastle had and the budget Birmingham City had, Chris has done fantastically wherever he's been.
"He's bubbly, he's intelligent and knows exactly what he wants and how his team should play. He was very well liked at Tottenham during the many years he spent there.
"Whoever comes in as manager, if it's Chris, has got a hell of a job to follow on from Paul Lambert. But Chris is very capable of handling that situation. He'd get on with it and cope with it very well."
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