Writing for The Telegraph, Carragher said: "Now Postecoglou has to make it count. It may yet become a 'Champions League or bust' shoot-out with Ruben Amorim if it is an all English Europa League final, but there is no doubt which manager's position is most perilous. Amorim is sure to get a chance to rebuild United over the summer whatever happens. Postecoglou is still not safe.
"That will surely change if Spurs go on to win their first major trophy for 17 years. In that situation, Daniel Levy and his board will have the same massive dilemma as Manchester United when Erik ten Hag won the FA Cup a year ago.
"After Ten Hag defeated Manchester City at Wembley, the United board made an emotional rather than logical decision. They were rightly criticised. All the evidence pointed to the downturn over the previous eight months continuing.
"If Postecoglou wins the Europa League, Levy will also have to choose between his head and heart. The case for keeping Postecoglou will be more compelling because, unlike United, Spurs do not have a long history of winning big prizes."