The controls mean banks and building societies will be required to check all applications for new current accounts with a specified anti-fraud organisation or data-matching authority.
Anyone who is in the UK illegally will not be allowed an account.
The Home Office said in a statement of intent that CIFAS would be chosen as the specified anti-fraud organisation to hold data on known illegal migrants.
The legislation continues the reform of the immigration system so that it is fair to British citizens and legitimate migrants.
Those with outstanding asylum applications or appeals will not be affected, nor will those who have been granted leave to be here, including refugees. Only data relating to known illegal migrants will be passed on.
CIFAS is a leading not-for-profit anti-fraud organisation with many banks and building societies as members. Over the last two years CIFAS and the Home Office have worked with UK financial service providers to avoid fraud amounting to £25million.