Passage of the Bill has been delayed for weeks as the Lords insist on key amendments, including exemption of Afghan service veterans from the scheme to remove migrants to Rwanda.
Conservative MPs and opposition parties are now urging Sunak to reconsider plans to relocate Afghans who served alongside British soldiers, including interpreters, to the East African country as part of the government's strategy to deter people from arriving in the UK via small boats across the English Channel.
Labour peer Des Browne, who was defence secretary from 2006 to 2008, told The Guardian that some of the MPs in government were uncomfortable with Sunak's tough stance on the matter.
'I can barely walk five yards in the corridor without some Conservatives stopping me and saying they wish the government would budge on this issue,' Lord Browne stated.
'They can't understand why the government couldn't concede something on this. I don't understand why they're not more questioning about the implications of this for our security and for our own armed forces.'
Lord Browne insisted that some of the Afghan veterans had resorted to using 'irregular' routes to enter the UK after being denied entry by the UK government.
Conservative MP for South Swindon, Robert Buckland, expressed his optimism that the Bill, which declares Rwanda a safe country, would pass today if ministers were willing to compromise.
'There is still a cohort of Afghans who are not in a safe place and who put themselves in the line of danger not just to secure their country, but in the interests of freedom,' he said.
'If the government were to offer some concession, as they did with modern-day slavery, it would unlock the bill.'
Sunak is keen on the MPs sitting through the night to pass the Bill after several setbacks. He expressed his confidence on Friday that the Bill would sail through.
'Repeatedly, everyone has tried to block us from getting this bill through, and yet again you saw this week Labour peers blocking us again, and that's enormously frustrating,' he said on Friday after the Lords insisted on amendments to the Bill on Wednesday.
'Everyone's patience with this has worn thin, mine certainly has. Our intention now is to get this done on Monday. No more prevarication, no more delay, we are going to get this done on Monday, and we will sit there and vote until it's done,' he added.
Besides the Afghan exemptions, the Lords are demanding tighter checks on the safety of Rwanda for asylum seekers and "due regard" for domestic and international law.
The scheme was first announced in April 2022 by former Prime Minister Boris Johnson's government following a deal with Rwandan authorities.
The Bill was introduced to Parliament after the UK Supreme Court ruled that the government's scheme was unlawful. The judges argued that genuine refugees being removed to Rwanda faced the risk of being returned to their home countries, where they could face harm.
The UK plans to offer failed asylum seekers up to £3,000 (Rwf4,866,727) to move to Rwanda under the new voluntary scheme.
Wycliffe Nyamasege