Home Secretary Yvette Cooper has told the BBC the level of "dangerous" small boat Channel crossings is "far too high" but she refused to set out a timetable to reduce the number.
More than 20,000 people have made the crossing to the UK since Labour took power, up on 17,020 during the same period last year.
It comes as the Home Office said it was on track to return the highest number of failed asylum seekers in five years, with nearly 13,500 people sent to their home countries since the election in July.
During a visit to Rome, Cooper told the BBC's Sunday with Laura Kuenssberg show there had been a "bad history" of home secretaries giving pledges on migration they then failed to keep.
Also speaking to Kuenssberg, shadow home secretary Chris Philp said the uptick in returns was part of a "trend" which started under the previous government.
He argued that Labour had made a "huge mistake" by scrapping the Tory government's Rwanda deportation scheme, which had yet to get off the ground by the time of July's election, two years after it was announced.
The number of migrants crossing in Labour's first five months in office is down on the same period in 2022, a record year for arrivals, but similar to the number who crossed in that period in 2021.
Cooper said the number of crossings was still "deeply damaging" and "dangerous", and was undermining border security as well as putting lives at risk.
"Of course we want to continue to progress, of course we want to see the boat crossings come down as rapidly as possible.
"What we are not going to do is deal with this by slogans. Rishi Sunak said he'd stop the boats in a year."
Asked whether the government's reluctance to set public targets on reducing the number of small boat crossings shows it is not a priority, Cooper said: "Quite the opposite.
"We've made clear that border security is actually one of the foundational issues".
Labour has previously expressed an interest in studying Italy's deal with Albania, under which some migrants rescued by the Italian coastguard will be sent there to have their asylum claims processed.
That five-year agreement has faced significant challenges, with transfers recently halted by a court in Rome.
Asked whether Labour would also consider processing asylum claims offshore, Cooper said the government would "look at whatever works" but emphasised that any such scheme would need to be "effective".
She also blamed an increase in hotel places for asylum seekers since the election on a "collapse in decision-making" ahead of the vote, which she argued had left Labour with a "soaring backlog" of cases on entering office.
Speaking on Sky News, Home Office minister Angela Eagle defended the lack of an official timetable for reducing crossings, adding that the public wanted to see "steady progress" rather than a "date snatched out of the air".
But Philp told the BBC Australia's offshore processing policies had shown that "deterrent works" - and pledged the Tories would aim to resurrect a Rwanda-style deal if they returned to power.
It was "telling", he argued, that reducing Channel crossings had not featured in the six government "milestones" laid out by Sir Keir Starmer last week.
The Conservatives under Kemi Badenoch have also separately committed to setting a cap on annual legal migration into the UK.
Asked where the cap would be set, Philp did not offer details beyond saying it would be "far, far lower" than the 350,000 predicted for future years by official economic forecasters.