The former Ireland international believes that the Italian can navigate Ireland to Brazil in 2014, but also name-checked three people he would like to see take over the role
Brady, who was formerly an assistant to Trapattoni, stressed, however, that Ireland still have a chance to qualify for the 2014 World Cup in Brazil and as such, he remains supportive of the current coaching regime.
"I've been vocal in his support because I don't think the group is over yet," Brady told reporters. "I still think we have a fair chance. I think the team is still playing for him. I don’t care what anyone says."
He continued: "Even with the whacking we got by Germany (a 6-1 loss at home), I think the team is still with him. That's why I've been very supportive. Let's see how the points tally ends up.
"But if we don’t qualify, probably Marco and Giovanni will know their contracts won't be renewed," he added.
The former Arsenal and Juventus midfielder named three managers that he would like to see assume the reins of the Boys in Green after Trapattoni's tenure, but warned that it would be every bit as costly as employing Giovanni Trapattoni.
"Trap is being beaten with this salary stick all the time but it is relevant. If you're going to get a top manager, you’ve got to be able to pay him," he said.
"The Premier League is getting wealthier all the time, and the Championship too. I would see it as three candidates, but would Chris Hughton leave the Premier League to come and manage Ireland? That's a big question mark."
He added: "Mick McCarthy is another. He's at a great club at Ipswich, who have had their problems, but Mick is relishing the challenge of sorting them out.
"The other guy I'd like to see is the guy who got the sack recently, Owen Coyle. I think he is a very good, experienced manager. He may be somebody the FAI would consider if Giovanni goes. But I'm hoping Giovanni pulls the ace out of the pack."
The Italian coach is, however, unlikely to step down voluntarily, according to Brady, who does not think that Trapattoni is inclined to quit from a role.
"I don't think it's in his make-up to quit," he added. "I think he'll have to be forced to step down, if he is going to step down."