The official news agency IRNA said lawmakers had included amendments requested by a powerful clerical body, which must vet all legislation passed through parliament before it becomes actual law.
Supporters hope the legislation, once passed, will allow Iran to join an international convention against the funding of terrorism and comply with measures against money laundering set by the Financial Action Task Force (FATF).
Hardliners in parliament have opposed parts of the legislation, saying it could hamper Iranian financial support for allies such as Lebanon’s Hezbollah, which the United States has classified as a terrorist group.
The Guardian Council clerical body had asked lawmakers to include clauses including a guarantee that the law would not clash with the constitution.
The Paris-based FATF said in October that Iran had until February to complete reforms that would bring it into line with global norms or face consequences.
Foreign businesses say legislation that includes FATF guidelines is essential if they are to increase investment.
Many Western companies have already ceased cooperation with Iran after the United States pulled out of a 2015 nuclear deal between world powers and Tehran in May and reimposed sanctions.