Ireland are currently without a permanent coach following the departure of Giovanni Trapattoni last month.
Brady, who won 72 caps for his country, now believes that current Ipswich Town boss McCarthy should return to the role after previously enjoying an six-year spell in charge from 1996 to 2002.
"For me it would be Mick," he told RTE Radio. "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'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."
Ireland finished their unsuccessful World Cup qualification campaign on Tuesday night with a 3-1 win over Kazakhstan.