Saudi Arabia’s turbocharged attempt to turn its domestic soccer league into one of the sport’s most glamorous has already attracted Cristiano Ronaldo, one of the greatest stars of his generation, and Karim Benzema, the reigning world player of the year. Those deals, though, pale into comparison with its most ambitious target yet: Kylian Mbappé.

Over the course of the weekend, one of the Saudi Professional League’s more prominent teams, Al Hilal, submitted an offer worth $332 million for the France striker to his current team, Paris St.-Germain. Should the deal go through, it would make Mbappé the most expensive player in the sport’s history by some distance, dwarfing the $263 million P.S.G. paid for the Brazilian forward Neymar six years ago.

The official bid was sent to P.S.G.’s chief executive, Nasser Al-Khelaifi, on Saturday. It was signed by Al Hilal’s chief executive, and it confirmed the price the club was prepared to pay and requested permission to discuss salary and the length of a contract with Mbappé. On Tuesday, it was reported by some news outlets that P.S.G. had granted that request.

Al Hilal was expecting to hold initial talks with Fayza Lamari, Mbappé’s agent and mother, early this week, according to three people with knowledge of the offer, who spoke on condition of anonymity because they were not authorized to discuss the details. It is likely that the club will have to commit hundreds of millions of dollars more in salary to persuade Mbappé, 24, who is regarded as the likely heir to Ronaldo and Lionel Messi as the finest player on the planet, to leave P.S.G. for a team in what was most recently ranked as soccer’s 58th strongest domestic league.

Mbappé is already lavishly remunerated at P.S.G., his hometown club. Last summer, he was handed a contract worth $36 million a year, complete with a $120 million golden handshake.

Even the amount of money that P.S.G.’s ultimate owner — Qatar Sports Investment, drawing on the wealth of the Qatari state — can afford to pay him, though, may not prove off-putting to his prospective employer: Al Hilal is now one of four Saudi teams majority owned by the Public Investment Fund, Saudi Arabia’s sovereign wealth fund.

There is an element of opportunism in Al Hilal’s approach. Mbappé’s future has been the subject of intense speculation since the start of June, when the player informed P.S.G. that he intended to see out the final year of his current deal and walk away as a free agent in 2024.

P.S.G. has insisted that it will not contemplate losing such a prized asset for nothing, informing Mbappé that he must sign a new contract — one that would extend his stay beyond 2024 — or face an uncertain future: either being sold or having to spend the season on the substitutes’ bench.

The club has sought legal advice to gauge the strength of its position. Mbappé has maintained that he intends to spend the coming season in Paris, although he was omitted from the squad for the club’s preseason tour of Asia last week as a result of the standoff.

Al Hilal is not the only team hoping to take advantage of the growing schism between P.S.G. and one of soccer’s most talented players and most marketable names.

P.S.G. has received several inquiries about Mbappé’s theoretical price tag. Chelsea, now owned by a consortium that includes Clearlake Capital Group, the private equity firm, has asked P.S.G. how much the player would cost. Barcelona, the Spanish champion, has discussed a deal in which more than one of its own prime assets would arrive in Paris in an exchange.

Real Madrid, long assumed to be Mbappé’s preferred destination, has yet to show its hand. Some executives at P.S.G. believe a deal is already in place in which Mbappé would move to the Spanish capital next summer.

It is that expectation that Al Hilal — most likely not the sort of place that Mbappé, at this stage of his career, would ordinarily have considered as his natural next step — hopes may provide it with an advantage.

It has been reported that, despite all the money it is prepared to spend to secure his arrival, the Saudi club would allow Mbappé to leave for Spain after just a season in the Middle East.

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