Peers were urged to “calm down” and allow the Government’s Rwanda deportation legislation to progress, as MPs voted to overturn amendments made by the Lords.

The Safety of Rwanda (Asylum and Immigration) Bill will not receive royal assent until both the House of Commons and House of Lords agree its final wording, a process known as parliamentary ping-pong.

MPs returned from the Easter recess on Monday to discuss six further changes made by peers, with the Government tabling motions to disagree with them – while also moving its own proposal in a bid to ease concerns over how the Bill operates in relation to modern slavery victims.

The Bill seeks to compel judges to regard the east African country as safe in a bid to clear the way to send asylum seekers who cross the Channel in small boats on a one-way flight to Rwanda.

The amendments overturned included an attempt by peers to ensure the Bill has “due regard” for domestic and international law and that Rwanda is only regarded as safe for as long as the provisions of the UK’s treaty with that country are in place.

Peers are expected to consider the Bill again on Tuesday and could make further changes which would prolong the parliamentary wrangling.

Speaking in the Commons, Conservative MP Sir Bill Cash (Stone) said: “The real question now is, let’s get this Bill done, let’s get the House of Lords to calm down a bit, let us also at the same time wait for what is inevitably going to be another claim and then see what the judgment of the Supreme Court is on the wording – providing it is clear and unambiguous – of this Bill.

“That is all I need to say, I may come back again however if there is another insistence by the Lords on these ridiculous amendments.”

For Labour, shadow Home Office minister Stephen Kinnock said the Rwanda scheme is “doomed to fail”.

He said: “The boats have kept coming, the backlog has kept growing, and the people smugglers are still laughing all the way to the bank. Two years of headline-chasing gimmicks, two years of pursuing a policy that is fundamentally unworkable, unaffordable and unlawful. Two years of flogging this dead horse.

“I am an inveterate optimist, so I truly believe that one day the benches opposite will come to understand that hard graft and common sense are always more effective than the sugar rush of a tabloid front page.”

Opening the debate, Home Office minister Michael Tomlinson said: “Here we are, back again, debating the same issues and amendments we have already rejected. We’re not quite at the point yet of completing each others’ sentences, but we are almost there.”

On amendment 10B, which sought to exempt agents, allies and employees of the UK overseas from being removed to Rwanda, he said: “This Government recognises the commitment and responsibility that comes with combat veterans, whether our own or those who showed courage by serving alongside us, we will not let them down.

“I want to reassure Parliament that once the United Kingdom’s special forces ARAP (Afghan relocations and assistance policy) review has concluded, the Government will consider and revisit how the Illegal Migration Act and removal under existing legislation will apply to those who are eligible as a result of the review, ensuring that these people receive the attention they deserve.”

MPs voted 312 to 253, majority 59, to reject Lords amendment 10B.

Sunday was the busiest day yet for Channel crossings so far this year after more than 500 migrants arrived in the UK in a single day.

The latest crossings took the provisional total for the year so far to 6,265 – 28% higher than this time last year (4,899) and 7% higher than the 5,828 recorded at this point in 2022.

PA news agency analysis of the figures suggests 75,629 migrants have made the journey since then home secretary Priti Patel signed what she called a “world-first” agreement in Rwanda’s capital Kigali on April 14 2022, with 43,328 taking place since Rishi Sunak became Prime Minister six months later.