Cockerill has had to learn his trade quickly after being thrust into Leicester's top job when South African Heyneke Meyer was forced to return home for personal reasons in 2009.
He was also acting head coach for a spell in 2007 before Argentinian Marcelo Loffreda took over for a short period.
"I've always wanted to work under an experienced head coach and to learn,'' Cockerill told the Daily Mail. "Unfortunately, by circumstance with what has happened here to Marcelo and Heyneke, I've never had the chance to work under an experienced coach.
"Perhaps, those days have gone. But if someone like Graham was available and I could use him as a sounding board and to pick his brains on things I would be a fool not to look at that. There's a lot of experience there and it would be a shame if it went to waste.''
Henry won't be short of offers. He became the bookies' favourite to replace former England coach Martin Johnson and was sounded out by the RFU but didn't want a fulltime international job.
Last night Henry was named 2011 New Zealand Rugby Coach of the Year to go with the 2011 IRB Coach of the Year award he won for the fifth time.