Cooper is back in action and played a pivitol role at the end of Queensland Reds Super Rugby campaign.
But he was not considered for the June Tests against Wales and with the likes of James O'Connor and Kurtley Beale also injured Barnes played a staring role.
Genia is delighted to have Cooper back but reckons Barnes deserves to stay at fly-half for the Rugby Championship kick-off.
"Quadey has done a lot of work in terms of getting his body right and I think he is fit, he is ready to go," said Genia.
"It's just a matter of whether you'd be willing, I guess, to put him into a Test match straight up and play 80 minutes against the All Blacks.
"I think that's more a call for (Wallabies coach) Robbie (Deans) than anyone else.
"From a player's point of view, it would be obviously great having him (Cooper) there, but Barnesy did a great job against Wales and in all fairness, I guess, probably deserves first crack at that No.10 jersey."
The live-wire scrum-half played every minute of the Reds' Super Rugby campaign and all of the Wallabies' summer Tests.
But after three weeks rest, Genia is feeling refreshed and is relishing the inaugural Rugby Championship.
Australia kick-off against New Zealand on August 18, and Genia has identified the breakdown as the key battle to win.
"I definitely appreciated the time off. I haven't lost any fitness, any conditioning, I've been working pretty hard," added Genia.
"They tend to dominate there which gives them good, clean quick ball and they tend to stifle a lot of opposition ball because they are so good in that area.
"We've identified that as a big focus in the last couple of weeks and leading into this game and that's an area where we're going to have to dominate if we are going to do well against them."