Real Madrid - the club widely thought to be leading the chase for the most potent forward in world football - are among a number of other clubs to have expressed an interest, the source said.
Al-Hilal have made a formal offer by letter, the source said, but such a transfer could only be made with the agreement of Mbappe who has never expressed an interest in moving to the fast-growing Saudi league.
"The club has given permission to Al Hilal to speak with Mbappe. That's the request of the letter and that's been granted," the PSG source said.
Riyadh-based Al-Hilal are one of the handful of clubs owned by Saudi Arabia's Public Investment Fund (PIF).
They have already signed central defender Kalidou Koulibaly from Chelsea and Ruben Neves from Wolves this summer.
Mbappe has one year left on his contract with PSG, who want him to leave now rather than for nothing next summer.
But Mbappe is refusing to sign a new contract and the French champions believe he has already struck a deal to move to Real Madrid next year for nothing.
Change in tactics
"In the group so far that have expressed their interest there are Manchester United, Inter Milan, Barcelona, Tottenham and obviously the Saudi club and Real Madrid," the source said.
The approach from Real appears to represent a change in tactics by the 13-time European champions, who tried to sign Mbappe last year before he agreed to stay with PSG alongside Neymar and the now-departed Lionel Messi.
It is widely thought that Real hoped to pick up Mbappe for free next year, attracting him with a colossal signing fee because they would not have to pay PSG anything for his services.
The standoff between Mbappe and the French club reached a new peak at the weekend when they dropped him from their squad to tour Japan and South Korea.
There have been suggestions that Mbappe would not be selected for an entire year if he fails to find a solution with PSG.
Mbappe, the club's all-time top scorer with 212 goals in 260 appearances, remains the second most expensive player in history behind his PSG teammate Neymar.
He was still a teenager when he joined PSG on a season-long loan from Monaco in August 2017.
The move turned into a permanent transfer the following year in a 180-million-euro deal.
That fee is exceeded only by Neymar, the Brazilian who cost PSG €222 million in 2017.
The third member of PSG's superstar trio, Messi, left at the end of last season to join MLS franchise Inter Miami.
The trio could not prevent PSG being knocked out of the Champions League by Bayern Munich at the last 16 stage last season although they went on to clinch the French title.