Proposed changes to EU law would force companies that transfer Bitcoin or other crypto-assets to collect details on the recipient and sender.
The proposals would make crypto-assets more traceable, the EU Commission said, and would help stop money-laundering and the financing of terrorism.
The new rules would also prohibit providing anonymous crypto-asset wallets.
The proposals could take two years to become law.
The Commission argued that crypto-asset transfers should be subject to the same anti-money-laundering rules as wire transfers.
"Given that virtual assets transfers are subject to similar money-laundering and terrorist-financing risks as wire funds transfers... it therefore appears logical to use the same legislative instrument to address these common issues," the Commission wrote.
While some crypto-asset service providers are already covered by anti-money-laundering rules, the new proposals would "extend these rules to the entire crypto-sector, obliging all service providers to conduct due diligence on their customers," the Commission explained.
Under the proposals, a company transferring crypto-assets for a customer would be obliged to include their name, address, date of birth and account number, and the name of the recipient.
David Gerard, author of Attack of the 50 Foot Blockchain, told the BBC: "This is just applying existing rules to crypto. This has been coming since 2019."
He said that although these were European proposals their impact would reach much further.
"If you want to make real money, you have to follow the rules of real money," he said.
To become law the proposals will need the agreement of member states and the European Parliament.