The 2018 World Cup winner, 23, had been heavily linked with a move to Real Madrid but has committed his future to the French champions.
Mbappe will sign a lucrative new contract with PSG.
The deal is still to be signed but PSG are likely to make an announcement around their final game of the season with Metz on Saturday evening.
According to Spanish football expert Guillem Balague, writing last month, PSG were prepared to offer him a 150m euro signing-on fee to keep him for another two seasons.
Mbappe had agreed a financial package with Real months ago for a potential switch to Spain, but has decided not to join Carlo Ancelotti’s side.
He initially joined PSG on a season-long loan from Monaco in August 2017 and has helped them win four Ligue 1 titles in five seasons.
The Qatari-owned side were desperate to keep him and were prepared to offer the player a £21m net annual salary.
Mbappe scored in each leg against Real in the Champions League last 16 this season, but it was not enough to prevent PSG from going out 3-2 on aggregate.
He is Ligue 1’s leading scorer with 25 goals and provided a further 17 assists in 34 games as PSG won the domestic title with four games to spare.
PSG boss Mauricio Pochettino had said on Friday he had no idea where Mbappe would play his club football next season.
“I hope Kylian is still here for many years to come but I also can’t lie. I don’t know what is going to happen,” said the former Tottenham boss.
“There are lots of rumours going around but the player is the one who will have to talk about this.”