The Argentina captain's current deal expires at the end of the season.
Sport says PSG have now contacted Messi to alert him to their contract plans. The Parisiens wish for the midfielder to remain for two more years.
PSG will offer a new two-year contract to 2025, though would be open to a proposal of one-year plus a 12 month option.
PSG are making their move in order to end all claims of Messi returning to Barca this summer.