Martinez has come under pressure due to Everton’s disappointing performances and results in the Premier League this season.
The latest defeat for the Toffees came on Wednesday evening, when they went down 3-0 to Sunderland away from home at the Stadium of Light.
Former Liverpool defender Carragher has suggested that if Everton do not want Martinez to be in charge of the first team next season, then they should sack him before this weekend’s final round of matches in the Premier League.
"If Bill Kenwright and the new owner who has come in alongside him want to change the manager at Everton Football Club for next season they've got to do it tomorrow,” the former England international told Sky Sports.
"Basically, they shouldn't be putting Roberto Martinez through the final game of the season at home to Norwich.”
"I think it could be a poisonous crowd and reaction to Roberto Martinez and the team at the weekend,” added Carragher. “If they want to keep the manager then keep him and he obviously has the final game and back him. But if they are thinking of changing the manager, don't let him go through the weekend.”
Everton’s final game of the 2015-16 Premier League campaign is against relegated Norwich City at Goodison Park on Sunday afternoon.
The Merseyside club’s fans are not happy with the Spaniard, who does not look to have any answer to his team’s current problems.
The Toffees are 12th in the Premier League table at the moment with 44 points from 37 matches.