The Ligue 1 champions' willingness to part ways with their superstar forward comes after sources confirmed multiple reports that Mbappe sent a letter to PSG informing them he would not exercise the option to extend his deal beyond 2024.
With PSG not wanting Mbappe to leave as a free agent after next season, sources said club chairman Nasser Al Khelaifi is now begrudgingly open to letting the 24-year-old leave Paris.
ESPN sources added that PSG would be looking for a fee in the region of €150 million ($161.5 million).
Two years ago, PSG turned down a bid in the region of €180m from Real Madrid, who have pursued Mbappe since he was 11 years old, for the World Cup-winning forward before he went on to sign his current contract.
Madrid has already agreed to a deal for England midfielder Jude Bellingham. The potential acquisition of Mbappe would be a major statement of intent after they relinquished their LaLiga and Champions League titles this year.
Mbappe had refused to discuss his PSG future following their Champions League round-of-16 defeat at the hands of Bayern Munich in March, another disappointing exit from Europe's top club competition.
There is also uncertainty about the future of another PSG star in Brazil international Neymar.
But the potential departure of the 24-year-old Mbappe would be the greatest loss to PSG, given that he is a national icon and widely considered one of the few players capable of taking over from Messi and Cristiano Ronaldo as the game's biggest star.
Mbappe had until July 31 to trigger a one-year extension to his contract. He has been at the club since 2017 after signing from AS Monaco in a transfer worth a reported $190 million.
PSG were planning a change of strategy after moves for the biggest stars failed to bring success in Europe. The new focus would be on young, French talent. Mbappe was expected to be at the heart of that shift and it had been hoped he would sign a longer-term contract.
Mbappe has won five French league titles with PSG and was a World Cup winner with France in 2018.
Information from The Associated Press was used in this story.