Proposed legislation that has not been introduced by the government must pass all its parliamentary stages before the end of each session, which usually last between one and two years.
If it runs out of time, the legislation typically fails.
Supporters now believe it is “effectively impossible” for the bill to legalise assisted dying in England and Wales to pass before the end of this session, expected in May.
This is largely because more than 1,200 amendments have been tabled in the House of Lords, principally from opponents to the legislation.
With only three of 14 days of debate in the Lords remaining, around half of the amendments are yet to be discussed.
Labour MP Jess Asato, who opposes the bill, said: “The sponsor of the bill has rejected 99% of suggested improvements and amendments in the House of Lords and so it still contains all the same faults and issues.”
She added: “Any MP that voted to push this bill through would do so knowing that it is unsafe and would harm vulnerable people.”
A source working with Labour opponents to the bill said: “For Labour MPs to continue to agitate for this unsafe, badly written and divisive bill rather than focus on the reasons people voted for us at the election is madness.
“Labour needs to work on delivering on our manifesto and running the country well, not rerunning contentious arguments about assisted dying.”
This comes as MSPs in Scotland prepare to vote on their own assisted dying bill on Tuesday, that would allow mentally competent adults with a terminal illness to seek medical help to end their lives.