In an interview with Sky Sports, Carragher said: "My feeling is that the whole point of international football, certainly with the major nations who compete for the tournaments, is that it's your best v their best. Club football is different and we love it, with different nationalities and different coaches in the Premier League or at the Champions League level. It's a real mix.
"The thing that makes international football different to the club game is it is people from your country. I know that's skewed a little bit with players who maybe aren't born in the country or it goes back to their grandparents. And I also understand certain nations who have a foreign coach who is going to have an influence on them that will drag the whole footballing structure up with them and pass on his experience and expertise to all coaches throughout the system and the country."
However, the former Liverpool defender believes that the FA should have continued to back English managers after Southgate's success.
"When I think of England, we are so close to winning a major tournament with so much good work that has gone into producing these players at St George's Park, it just doesn't feel right to me that we have a foreign coach," he said.
"It's not just about England - I don't think Italy should, I don't think Germany should, I don't think France should. Portugal have it at the moment with Roberto Martinez which I find strange. That's my take on it. I'm not bothered either way. I must be perfectly honest, I'm not the most patriotic of people. It's not just about 'the England manager has to be this'. I'm not going to lose any sleep over it. England have got a brilliant manager but I think England should have an English manager."