The report stated that the Argentine great's father Jorge has been negotiating with PSG director of football Luis Campos for months, and the agreement came after a meeting with the club’s Qatari owner Nasser Al Khelaifi in Doha.
Messi’s current contract ends in June 2023.
Messi was the player of the tournament in Qatar as he guided Argentina to a thrilling win over France at the Lusail Stadium.
Barcelona president Joan Laporta had recently expressed interest in bringing the seven-time Ballon d’Or player back to the club, stating: "At the moment he is at PSG and we would very much love him to come back one day."
Messi joined PSG in 2021 on a two-season deal with an option for another year after having spent his entire professional career with Barcelona.
He collected four Champions League titles in 2006, 2009, 2011 and 2015 in Spain as well as 10 La Liga crowns.
Argentina's World Cup triumph saw them move up to second in the Fifa rankings with quarter-finalists Brazil still top of the table.
The Argentines moved up one place, as did runners-up France, who are now third, with Belgium dropping to fourth.