The extra figure is believed to run into the tens of millions of pounds, although the final sum has not been revealed by the Home Office.
Since 2015 the British taxpayer has handed over £124million to France.
Home Secretary Amber Rudd met her counterpart Gerard Collomb in London last week and French officials said that Ms Rudd was prepared to meet their demands for more money but there has been no new agreeement as yet.
Ministers are also said to be looking to renew the security contract surrounding operations at Eurotunnel terminals at Coquilles, Calais and Dunkirk.
This could cost around £80m, although the Home Office believes the figure could be lower.
Emmanuel Macron during his presidential campaign threatened to tear up the Le Touquet treaty, which permits British police to check and stop migrants in Calais.
However, Mr Macron’s administration now wants to reform the agreement with Britain paying for tighter security.
A Home Office spokesman said: “We work closely with the French authorities at all levels to reduce migrant pressures and target criminal gangs involved in people trafficking, both in northern France and further afield in source and transit countries.
“The Home Secretary’s meeting with the French Interior Minister last week is part of the ongoing dialogue to continue the work set out in the UK-France Declaration of 2015.”
People-traffickers have recently started using a new tactic as a way to evade detection, a French official has said.
A “saturation” technique involves the traffickers sending up to six vans to a French port all at the same time in the hope that while one might get stopped, the others will make it to Britain.
Hundreds of refugees and migrants are believed to be living in the French port still, just over a year since the camp – known as the “Jungle” – was razed by police.
The region’s conservative mayor Natacha Bouchart claimed last month up to 1,000 migrants are sleeping rough on the city’s streets or in secret makeshift camps on the outskirts of town.
She said: “Between 800 and 1,000 people have returned to Calais. Migrants have decided, in an anarchic manner, that they have the right to be here. But don’t let them fool you – they are not helpless. Migrants know Calais better than anyone else.
“They have smartphones and nice clothes. They’ve been told that they have rights, but no duties.
“They drink themselves senseless – they down litres of vodka – and get into fights.”