It must complete all its parliamentary stages before the next King’s Speech, which the BBC revealed is expected in early May.
If it does not pass all its hurdles before then, the bill will fall.
However those behind the bill believe rarely used powers to limit the Lords’ ability to block legislation could bring the bill back for a second time.
The Parliament Acts mean that a bill can become law without the approval of peers, if it is rejected by the House of Lords in two consecutive parliamentary sessions.
However, this would rely on either the government offering its own parliamentary time for the bill, or an MP willing to bring an identical bill being drawn near the top of a ballot for Private Members Bills at the start of the new session.
The effect would be to delay the bill’s passage into law until 2027.
A source close to Kim Leadbeater – the backbench MP who introduced the bill – said: “This issue has to be resolved. The time has come for Parliament to decide its view. It is far better for that to be now than we have to go through it all again.”
Thursday’s motion, tabled by the bill’s lead backer Lord Charlie Falconer, says that “in order to allow the House to complete its scrutiny of the Terminally Ill Adults (End of Life) Bill and return it to the Commons in reasonable time before the end of the current parliamentary session, further time should be provided for consideration of the Bill.”
Supporters believe it will put pressure on peers to speed up their debates. But many, including the incoming Archbishop of Canterbury, remain implacably opposed.