Ireland recently sacked Giovanni Trapattoni following their unsuccessful World Cup qualifying campaign which left them fourth in Group C, six points behind runners-up Sweden and 14 adrift of Germany.
McCarthy, currently in charge of Ipswich, led Ireland to the 2002 World Cup finals where he infamously fell out with skipper Roy Keane.
Although Keane flew home before the tournament started, the squad still qualified for the latter stages thanks largely to a draw with Germany and they were eventually knocked out by Spain following a penalty shoot-out.
Earlier this week Ireland captain Robbie Keane called for McCarthy to be considered as Trapattoni's replacement and former star Brady told RTE Radio: "For me, it would be Mick.
"I don't know what his contractual situation is at Ipswich but what I do know is that he's demonstrated before that he's capable of doing the job.
"He has broad shoulders, he had to put up with an awful lot of messing when people turned against him.
"Most of the media turned against him, public opinion turned against him because of how popular Roy Keane was. But the man got on with it.
"He's demonstrated since then he's a very, very capable manager. I don't know if it's possible to get him but he would be number one on my list."
Other leading contenders include Martin O'Neill and Roy Keane, but Brady thinks the new boss will inherit a current crop of players that is not as good as in previous eras.
He added: "Robbie Keane himself, Richard Dunne are coming to the end of their careers. These are guys who were big, big players for the Irish team.
"The lads replacing them are not as good. That's the reality of the situation. I think Robbie is talking about a man with broad shoulders. That's what we need.
"We all expected that Martin O'Neill would be the next manager but it hasn't happened.
"He probably doesn't fancy the job for the reasons I've already stated. The expectation of the situation is too exaggerated."