A wavering group of MPs who backed parliament’s historic vote in favour of assisted dying may yet oppose its passage into law without further reassurances, the bill’s supporters are being warned, amid concerns that significant hurdles still remain.

MPs voted in favour of a change with a 55-vote majority on Friday, after a momentous five-hour debate over allowing assisted dying for terminally ill adults in England and Wales with less than six months to live.

However, Kim Leadbeater, the MP behind the legislation, is being urged to shore up support by immediately asking all those who backed her bill about their remaining concerns. The Observer has spoken to Tory and Labour MPs who backed the bill, but are reserving judgment before the final vote next year.

David Davis, the Tory former Brexit secretary, is among the group. He told the Observer that a proactive approach from the leading advocates of the bill could ensure any remaining doubts were overcome. “This does not have an overwhelming majority,” he said. “The second reading of the [1967] abortion bill went through by 223 to 29. I think a fair number of MPs voted on the premise I was making – that this is a bill which can be properly modified to make it right. We also had about 30 abstentions.

“If 30 MPs changed their mind, it would go the other way. So if I were advising Kim Leadbeater, I would write to every single person who voted for the bill and say, ‘What are your concerns?’ Address it head-on.”

Concerns among waverers are focused on the safeguards around Leadbeater’s terminally ill adults (end of life) bill, as well as the risks of coercion. Others want doctors who approve the necessary medication to have a special licence. Under the terms of the proposed law, two doctors and a high court judge have to sign off any request for assisted dying. Additionally, Davis and others are suggesting further attempts to make it harder to extend the scope of the bill in the future.

An ally of Leadbeater said that the next phase of the bill, which will see it discussed and scrutinised by a cross-party group of MPs, would ensure all concerns were addressed. “All MPs will have the opportunity to feed in their views on how the bill can be improved,” they said.

“Kim will be listening hard to what they say and is very aware of the need for the committee to reflect opinion across the House and to take MPs with her over the next weeks and months. So those MPs who said in the debate that the bill wouldn’t be subject to real scrutiny and significant amendments will be proved wrong.”

Some Labour MPs also have political concerns about the bill. They said it risked being a significant distraction to Keir Starmer as he attempts to show his government is making a tangible difference to people’s lives. “This is going to take up a huge amount of time,” said a Labour MP who voted against the bill. “I find that a bit frustrating.”

Figures from across the Commons are now demanding that the bill be given as much time as it needs on the floor of the house next year. They pointed to the extensive time given to the 1967 Abortion Act originally proposed by the Liberal MP David Steel.

Meanwhile, Whitehall is beginning the work of drawing up impact assessments covering the proposals. There are concerns in government about the impact on the NHS. Health secretary Wes Streeting, who opposes the bill, has said it will have “resource implications” for the health service. Shabana Mahmood, the justice secretary, is also fiercely opposed.

Parliamentary sources said that the bill was likely to be studied by a committee of MPs from January until March, appearing in the Commons in late April at the earliest. Unusually, the committee will take oral and written evidence. MPs also want more time in the subsequent Commons debates on the issue.

Support in the House of Commons reflects the views of the public, according to the latest Opinium poll for the Observer. Almost two-thirds (64%) support making it legal for someone to seek “assisted dying” in the UK, with 19% opposed. Cross-party support is high, with 60% or more of voters from all five of the largest parties supporting new legislation.

The poll found that the best argument for a change in the law, according to the public, was that it might “offer dignity and relief for terminally ill people at the end of their life”. Some 57% saw this as the strongest argument, while 37% thought it was “offering people the right to choose how they die”. The biggest concern was that “vulnerable people might feel pressured to choose assisted dying”, with 40% identifying this as the strongest argument against.