Priti Patel is to hold meetings with the French interior minister, Gerald Darmanin, after a tense exchange between the two countries over the number of migrants crossing the Channel to the UK.

The talks on Wednesday follow an influx of small boats during warm weather this week, increasing pressure on the home secretary from Downing Street and other members of her party.

The Home Office said 785 people crossed the Channel in small boats on Monday, short of last month’s record daily total of 828 migrants.

Patel started a war of words on Monday when she told MPs that Britain could withhold GBP54.2m it had promised to pay France to help deal with the situation, unless more boats were intercepted.

Conservative MPs have called for the home secretary to break international law and send all migrants arriving illegally by boat straight back to France.

A detailed rebuttal from the French interior ministry on Tuesday warned the UK not to attempt any action that was contrary to international law, saying there would be “consequences” if Britain refused to hand over cash that was agreed in July to combat small boat crossings.

The ministry said the deal was negotiated “in detail with the British side” and there was “never any question of making payment conditional on targets … Such an approach would reflect a serious loss of confidence in our cooperation.”

A record 13,500 migrants have crossed the Channel in small boats this year, including 1,000 in the past two days. Two hundred were prevented from crossing by the French on Monday, when 742 reached the UK.

The French ministry said it had prevented more than one in two crossings since January, or more than 10,000 migrants, after deploying “considerable and constantly increasing resources”. It had also doubled its resources in southern France to tackle migrants crossing the Mediterranean.

Pierre-Henri Dumont, France’s MP for Calais, accused Patel of attempting to use British taxpayers’ money to pile pressure on the French authorities.

“We are just doing our jobs and trying to save lives in the Channel and make sure that it does not turn into a graveyard,” he said. “The UK needs to address the causes that make people want to claim asylum in the UK. Many of these people are from former British colonies.”

Conservative MPs who lobbied Patel on Monday to send boats straight back to France have been criticised by refugee campaigners.

Minnie Rahman, the campaigns director at the Joint Council for the Welfare of Immigrants, said: “What MPs are suggesting is that the government take illegal action by sending people to France without first processing their claims. Hysterical, dog-whistle responses do nothing to help asylum seekers get the protection that they need.

“The only way to end these journeys is for the government to share responsibility with France, and ensure that people are able to travel to the UK safely.”