Trapattoni was sacked on Wednesday, the day after the Republic's 1-0 defeat against Austria which leaves their hopes of qualifying for the 2014 World Cup in Brazil hanging in the balance.
And the former Republic of Ireland duo, who have amassed 157 caps between them, would be happy to see Martin O'Neill step in and replace the Italian.
"He's got more experience than anybody out there so it wouldn't be a bad shout," Staunton told Sky Sports News.
"He's managed Celtic for five years and done really well up there, and at other clubs he's done really well, so that wouldn't be a bad choice. Who knows what the FAI have in mind, but Martin would be up there high on the list.
"I would like to see an Irishman get it or somebody who's from these islands, who knows what the Irish boys are about and knows the mentality. The FAI has a tough job now."
McAteer added: "Martin O'Neill, with his man management skills, would be an ideal candidate."
The former Liverpool and Bolton midfielder paid tribute to the 74-year-old Trapattoni but insisted it is crucial for the FAI to make the right managerial appointment for the national side
"Giovanni is a lovely fella, very shrewd. Two poor results cost them qualification and it was time to move on.
"There are some good players coming through. It's important to get the right man now to bring those players through."
Another former Republic of Ireland international, Ray Houghton, added: "There is no doubt Giovanni did a great job with them.
"Maybe the last few matches things haven't quite gone the way he would have liked.
"Maybe one or two of the players lost faith in what was going on, but whoever it is he has got some young players to work with and I think a good group of players to work with.
"We just have to make sure we get the right one to come in and move the international team forward."