Dame Esther, who has been calling for a change in the law, said she was “thrilled”, adding: “I never thought I might live to see the current cruel law change.
“But even if it is too late for me, I know thousands of terminally ill patients and their families will be given new hope.”
However, Baroness Grey-Thompson told the BBC she was opposed over concerns “about the impact on vulnerable people, on disabled people, coercive control, and the ability of doctors to make a six-month diagnosis – but also the time and capacity they have to make sure it’s someone’s settled wish”.
Assisted suicide – intentionally helping another person to end their life – is currently banned in England, Wales and Northern Ireland, with a maximum prison sentence of 14 years.
Assisted dying is generally used to describe a situation where someone who is terminally ill seeks medical help to obtain lethal drugs which they administer themselves.
As a backbench MP, Leadbeater would not normally be allotted time for a full debate and vote in Parliament on one of her proposed bills.
However, earlier this year she came first in the private members’ ballot, meaning she will be given some of the limited time available for backbench MPs’ bills.
Speaking to the BBC, the Spen Valley MP said topping the ballot had prompted her to look into the subject of assisted dying “in a lot more detail” and that she believed there was a “real appetite” for MPs to have a debate.
“The current situation isn’t particularly safe and there isn’t actually the choice that I believe people deserve and should have,” she said.
“At the moment… you’ve got three options.
“You can suffer and have a very painful, very difficult death which is very hard for you and your family.
“You can potentially go to Switzerland, to Dignitas – you can only do that if you have lots of money and if you are fit and well enough to travel.
“The other option is you can take your own life… The trauma that that leaves families in is just heart-breaking.”
Leadbeater said she understood her fellow MPs would be wary of discussing the “emotive” topic.
“They are nervous in some respects, as am I… but I think the consensus is that the time is right to have that debate and discussion,” she said.
“I really hope that for my part, I can facilitate that in what is a respectful and compassionate way.”