Britain's Crown Dependencies have objected to moves to force them to be more open about who owns assets held there.
A group of MPs want Jersey, Guernsey and the Isle of Man to create public registers of beneficial ownership.
Tory MP Andrew Mitchell and Labour's Margaret Hodge are behind one renewed move.
The efforts were "contrary" to established constitutional relationships, the islands said.
The UK created its own publicly accessible beneficial ownership register in 2016, and an amendment forced the Overseas Territories to follow suit by 2020, later extended to 2023.
More than 40 MPs have signed an amendment to the Financial Services Bill requiring the UK to help all overseas territories and Crown dependencies set up the registers within their jurisdictions.
This, they hope, will shine a light on money laundering and the proceeds of people-trafficking and other crimes.
If passed next week, it would oblige them to do so by the end of 2020.
The laws would be "wholly unnecessary" as the governments were committed to exchanging ownership information with the UK to combat tax evasion, money laundering and corruption, they added.
"The standards applied by our jurisdictions exceed those of the United Kingdom; the registers in the Crown Dependencies contain information that is accurate and up to date."