Deschamps will leave the Les Bleus job after next summer's World Cup.
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Deschamps announced back in January that he would step down as coach. Since then, he has been linked with Saudi Pro League clubs.
Now Sky journalist Sacha Tavolieri states that the Saudi FA is stepping up their push for the Frenchman.
The FA is prioritising a move for Deschamps with a clear project being put together.
They want to utilise his vast experience to build a long-term national team with the goal of a competitive home World Cup in 2034.
Currently in charge of the Saudi national team is Herve Renard, with a deal to 2027.