Liverpool meet West Ham in London on Sunday on the back of nine defeats in their past 12 games.
Slot is under huge pressure going into the game and Carragher said in his column for the Telegraph: "Arne Slot has a week to save his job
"It is hard to believe that sentence is being written, but Liverpool's next three games are against West Ham United, Sunderland and Leeds United. Anything fewer than seven points will make an already unacceptable situation untenable."
Carragher hammered Slot for last season's title celebrations.
He also opined: "The immediate response to the victory over Tottenham Hotspur on April 27 was justified and understandable. The supporters were always going to start the biggest party for 35 years after being able to witness their side become champions.
"But for the manager and players, there were still four games left. No one wants to sound like a killjoy, but the time to arrange parties in Ibiza and Dubai was after the final game against Crystal Palace, not before.
"It may have seemed good fun seeing all the party videos and photographs on social media, but as the team failed to win any of their final four games, losing twice, accusations of it being a bad look had merit.
"This was the club’s 20th league championship, taking them level with Manchester United. Even without league titles, Liverpool have been a winning machine. They have won 20 trophies in the past 24 years because no sooner is one more honour on the board, the work begins to get the next.
"At best, the downing of tools before the season was over was wilfully unprofessional. At worst, Liverpool behaved like a small club winning the title for the first time."
He added, "Signing two expensive No 9s was wrong, too. It is like betting on two horses in the same race. You do not buy a striker for £80m and then two weeks later another for £125m, because if they are competing for the same position, only one can win. It cannot work."