The Parliament Act allows for a bill that has been passed by the Commons but rejected by the Lords to return in a new parliamentary session.
If an identical bill passes the Commons a second time, the Lords cannot block it again and the legislation will become law at the end of that second session even without the Lords’ approval.
The powers have only been used seven times since 1911.
There are also several hurdles supporters would need to overcome.
Someone willing to bring the exact same bill would need to be drawn high up in the ballot of MPs able to bring a Private Members Bill.
Asked if it was now impossible for the bill to pass, Lord Falconer told BBC News: “It’s very very difficult, it’s not impossible if the Lords were to change the way that they were dealing with it.”
“I’ve seen no sign so far that there’s going to be a change,” he added. “But if it goes on like this it has absolutely no hope whatsoever of getting out of the Lords.”
Pressed on the controversy of using the Parliament Act to prevent the Lords from blocking the bill a second time, Lord Falconer said: “The issue about assisted dying is very controversial, but ultimately somebody in our constitution has got to decide whether the country should make the change.
“The people who should decide it should be the elected representatives in the Commons. If they make up their mind but are blocked in giving effect to that decision by a small number of peers then the constitutional answer is the Parliament Act.”
Lord Falconer has written to all peers on Wednesday evening setting out a number of amendments he will table, aimed at addressing concerns such as around those with eating disorders becoming eligible for an assisted death, and toughening restrictions on advertising for the service.
“It is our responsibility as a House to find a way forward,” he told peers, urging them to back the changes.
However several peers believe the change is dangerous and could mean vulnerable people are pressured into ending their lives prematurely.
A source close to Labour MPs and Peers opposed to the Bill told BBC News: “Threats to use the ‘nuclear option’ of the Parliament Act to recklessly force through this bill, which poses such risks to the vulnerable, is the act of a bully who knows they are losing the argument on the substance…
“People need to be very clear, using the Parliament act to force this through would mean that none of the known issues with the Bill would be fixed.
“Every MP who voted to force it though would bear responsibility for the inevitable suffering and deaths of vulnerable people.”