LONDON — Britain took a crucial step on Friday toward making all adults presumed organ donors unless they say otherwise, which would add the country to a growing list of those that have adopted the policy to address a chronic shortage for transplants.

The House of Commons, on a sparsely attended voice vote, gave unanimous approval to send an organ donor bill to committee, where a final version would be created. Though it still could face procedural obstacles, it has the support of a rare alliance of the Conservative government, the leadership of the opposition Labour Party, and the British medical establishment, indicating that chances of passage are good.

“I’ve seldom seen such a unanimous range of support,” said Geoffrey Robinson, the Labour lawmaker who sponsored the bill and who was one of dozens from both major parties who spoke in favor of it on Friday. “This will save lives.”

The change would apply in England. The Scottish Parliament is also considering moving to an opt-out system, while Northern Ireland requires explicit consent. Wales has had presumed consent to organ donation since 2015.