The news was reported by Tunis Afrique Presse on Tuesday, after Tunisia suffered a 5-1 defeat by Sweden in their first Group F fixture in Monterrey on Sunday.
Lamouchi, the first coaching casualty of the tournament, was appointed in January on a contract until 2028 and paid the price for a dire performance.
He leaves his post with matches to come against Japan on Saturday and the Netherlands on June 25.
Renard, twice an Africa Cup of Nations winner, coached Saudi Arabia at the 2022 World Cup in Qatar and led them to a famous victory over eventual winners Argentina in the group stage.
He also coached France at the Women's World Cup in Australia where the team went down to the Matildas in the quarter-finals.
Earlier, the Tunisian Football Federation appeared to have announced Lamouchi's dismissal on their Instagram account. Written in Arabic, the statement was translated by Google.
"An agreement has been officially reached to dismiss coach Sabri Lamouchi," the statement read.
"Plans are underway to appoint Mondher Kebaier as the national team coach (on an interim basis) to complete the World Cup qualifying matches."
Kebaier, 56, is already in Mexico and led the side between 2019 and 2022.
However, it was subsequently reported the Tunisian Football Association were holding an emergency meeting, and Lamouchi had taken training.
"We have a problem with the coach," a Tunisian Football Association official told ESPN.
Tunisia are scheduled to face Japan in the same stadium on Saturday night before finishing their group against the Netherlands.
Lamouchi, 54, had already been under fire following a 5-0 loss to Belgium in a pre-tournament friendly. He has overseen one victory in five games, a 1-0 win over Haiti.
Tunisian media have reported there was a lack of harmony between the team's experienced players and newcomers. The team's Africa Cup of Nations campaign last winter ended with a disappointing defeat to Mali in the round of 16.
The Frenchman previously managed the Ivory Coast squad at the 2014 FIFA World Cup but failed to advance beyond the group stage.
In 1998, Tunisia sacked Henryk Kasperczak after two group games, both losses. They drew the third match but still came bottom of the group.
with DPA
