Sky News reported that Barclays, HSBC, Lloyds, RBS and Santander have agreed to pay money into the Big Society Bank.
The story says the banks could pay between £1bn and £1.5bn into the bank, which is being set up to provide capital for voluntary organisations that want to run public services.
It says the agreement could "buy pledges from the government" for the banks, including a deal not to introduce further taxes on banks' activities.
A report in The Guardian today also says high-street bank bosses are considering whether to pay more money into the Big Society Bank.
Some banks are "hoping to buy off any attempts by the government to impose new taxes or bonus rules on their sector" by funding the project, the report claims.
A statement from the Cabinet Office said: "As with all major industries, we are in discussions with the banking sector on a number of topics, but it would be inappropriate to provide a running commentary of our conversations."