According to Bloomberg, which cited unnamed persons familiar with the matter, both HSBC and Lloyds Banking Group are planning on agreeing to the code.
The two lenders are among a number of major institutions which have yet to commit to the governmentâs proposals.
In an interview with the BBC, chancellor of the exchequer George Osborne, revealed that only four of the 15 biggest banks in the UK have signed up to the code so far.
He said: âWe are going to be looking at the code of practice that the banks were supposed to sign up to make them good taxpayers.
âAt the best of times tax evasion is unacceptable. At a time like this it is immoral.â
Banks will now be expected to agree to the changes by the beginning of November of this year.
The legislation was proposed by Alistair Darling, chancellor with the previous Labour government, who claimed it would ensure that banks adhered to the âspiritâ as well as the âletterâ of the law.
Meanwhile, new research from the Trades Union Congress showed that banks will avoid paying Â£19 billion in tax on future profits.
Institutions will be able to make the saving by offsetting losses suffered during the financial crisis against their tax bills.
The trade union claimed in its report that the figure was the equivalent to more than Â£1,100 for every family in the UK.
By Jim Ottewill