The Labour leader made the pledge during a phone call to Dame Esther Rantzen, in a major victory for the Daily Express's Give Us Our Last Right campaign.
Childline founder Dame Esther, 83, has reinvigorated the battle for greater choice at the end of life since revealing her stage four lung cancer diagnosis and decision to register with Dignitas.
Sir Keir told her: "I'm personally in favour of changing the law. I think we need to make time.
"We will make the commitment. Esther, I can give you that commitment right now."
Speaking to the Express afterward, Dame Esther said Sir Keir had raised concerns about assisted dying becoming a "political football", with other parties feeling obliged to oppose it if Labour makes a manifesto commitment on holding a debate.
But his personal commitment was "crystal clear", she said.
She added: "He told me that when he was director of public prosecutions, every single case where assisted dying was suspected was referred to him to decide if the loved ones left behind should be prosecuted. So he has heard the stories of terrible suffering imposed by the current law.
"And he added compassion to the criteria. But he admitted that was not enough to protect loving, compassionate families from a police investigation, which is the danger if I have to go to Dignitas and want my wonderful family to come with me to say goodbye."
Dame Esther implored MPs who had not personally experienced the painful death of a loved one to listen to the wealth of heartbreaking stories and evidence from around the world.
She added: "Obviously there must be careful precautions in any new law to protect the vulnerable. Sir Keir is fully committed to that.
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