The Le Touquet accord stipulates that UK border controls will be conducted on French soil in the event of Channel crossings.
However Emmanuel Macron warned during his election campaign that under his presidency, France would no longer act as Britain’s “coast guard” and said he wanted to renegotiate the agreement.
Sources in Paris claim the British Government has agreed the treaty, signed in 2003, will not continue in its present form.
A source told the Telegraph the details were still being discussed, but “in principle, the British have accepted an additional protocol, a clause or a new treaty.”
The deal in its current form is controversial in France as it lands the French government with responsibility for illegal immigration into Britain.
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It means that illegal immigrants in France are barred from Britain before they arrive. Those found to have crossed the Channel illegally are simply returned to France.
According to Le Monde, Mr Macron will demand that Britain takes more migrants in order to guarantee his approval for a post-Brexit EU trade deal.
The Elysée confirmed on Monday that Mr Macron wants Mrs May to “speed up the transfer of migrants to Britain with legitimate reasons to go there such as family ties” and to “accept more unaccompanied minors and faster”.
Mrs May is also likely to agree to increase financial contributions, sources close to French interior minister Gerard Collomb claim.
A source claimed: “Our understanding is that they will pay more. The question is how much and for what.
“We have let them know of our needs and a figure, we’re talking tens of millions of euros.”
Britain is expected to agree to a joint force to process asylum applications on French territory.
Home Office and French interior ministry agents will jointly handle requests and should have to process them more quickly.
Mr Macron and Mrs May will discuss the issue at a bilateral summit in Sandhurst on Thursday.
The French President is expected to use his rising international status to strike a hard bargain with the Prime Minister.