Cooper was omitted from Robbie Deans' recent 30-man training squad, heightening speculation that he will not be required for the three-Test series with Warren Gatland's tourists in June and July of this year.
But Genia insists Cooper is the man he wants to share half-back responsibilities with against Britain and Ireland's elite, ahead of the likes of James O'Connor, Christian Lealiifano, Matt Toomua, Berrick Barnes and Kurtley Beale.
"I love playing with Quade. I don't feel as comfortable as I do with him," Genia said.
"We've got a long relationship. We've played together for a long time - I really enjoy that combination.
"More than anything it is a trust thing. In whatever decision each makes we back and support each other."
Cooper endured a tumultuous year with the Wallabies in 2012 and was fined for his comments about the "toxic" international environment under Deans.
The mercurial play-maker would love to face the Lions but he has played down any speculation and remained at his diplomatic best.
"It all depends on what the coach and the selectors are looking for," said Cooper.
"Personally I am doing my best to work with Will, get the best out of each other and the Reds - I am not looking further than that."
Australia will name their squad on May 19, before the Lions begin their 10-match tour against the Barbarians in Hong Kong on June 1.