The concession came as figures showed the number of migrants arriving in the UK after making the journey has topped 23,000 for the year so far, with more than 2,000 arrivals recorded last week alone.
The Prime Minister’s official spokesman said: “I think we’re in no doubt on both sides there is certainly more we can do.
“Obviously, we are in touch with our French counterparts about taking further action to prevent crossings.
“I think it is still worth noting that, for the first time ever since these sorts of crossings began, the numbers are tracking lower than they did the previous year. That is notable and significant.
The Prime Minister and Home Secretary should spare us the empty slogans and start working on solutions
“But, obviously, there is much more to do to get this situation under control. So, we will continue to work very, very closely with our French counterparts to up interceptions, both at the channel but further upstream as well.”
Crossings continued on Monday, marking the tenth consecutive day of arrivals amid a spell of warm weather.
Some 2,017 people were recorded making the journey between Monday September 4 and Sunday September 10, according to PA news agency analysis of Home Office data.
More than 800 arrived over the weekend alone.
On Saturday, 425 people crossed in seven boats, suggesting an average of about 61 people per vessel.
The average rose to around 65 on Sunday, when 389 people were detected in six boats.
The latest crossings take the provisional total for the year so far to 23,103. This is still lower than this time last year, when around 27,000 had already been recorded.
However, more than 3,000 have crossed since the start of September, compared to around 2,600 for the first 10 days of the month in 2022.
Around 30,700 crossings have been recorded since Rishi Sunak became Prime Minister on October 25 last year, official Government figures for the period and provisional data suggests.
They indicate about 40,700 crossings have been recorded during Suella Braverman’s time in office, taking into account the period including her brief first tenure between September 6 and October 19 2022, and since she was reappointed on October 25.
More than 63,000 people have made the journey since then home secretary Priti Patel signed the deal to send migrants to Rwanda in April last year, the figures also suggest.
Labour’s shadow home secretary Yvette Cooper said: “Less than two weeks ago, Rishi Sunak told the media that his plan to stop the small boats was working and that the British people should have confidence in him to deliver.”
She accused Ms Braverman of making “countless claims and promises”, adding: “The Prime Minister and Home Secretary should spare us the empty slogans and start working on solutions.”
A Home Office spokesman said: “The unacceptable number of people risking their lives by making these dangerous crossings is placing an unprecedented strain on our asylum system.
“Our priority is to stop the boats and our small boats operational command is working alongside our French partners and other agencies to disrupt the people smugglers.
“The Government is going even further through our Illegal Migration Act which will mean that people arriving in the UK illegally are detained and promptly removed to their country of origin or a safe third country.”