According to reports, Al-Nassr and Al-Ittihad are interested in a move for the player and could make a salary bid worth €100m inclusive of the image rights and commercial deal.
Transfer journalist Fabrizio Romano claims Saudi emissaries are now in London to present an official proposal to Kante and his representatives.
The Chelsea midfielder is out of contract this summer, meaning that he'll be available on a free transfer at the end of June.
The 32-year-old is widely expected to agree a new deal with the Blues with the hopes of being a key player under Mauricio Pochettino.
Kante missed long spells of the season due to a hamstring injury, but came back into contention in the final weeks of the campaign.
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Asked if he'd like to stay with Chelsea, Kante told Sky Sports last month: "For sure.
"It's an exciting project for the club. Unfortunately this season hasn't been to Chelsea's standards.
"But everyone wants to go in this way, the way of success and winning titles.
"I hope this club will again fight in this direction.
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"Let's see where I will be, but I hope that this is a change for the club, that is most important."