A senior banking source said the offer, which compares with an estimated 60bn the major banks have lent to SMEs in 2010, was made at a meeting with Chancellor George Osborne and Business Secretary Vince Cable at the Treasury yesterday.
It is believed about 200bn would be made available to business customers, including larger customers, in total during 2011.
A Department of Business spokesman said after the one and a half hour meeting that lending to small businesses and "pay discipline" were two key topics discussed. Attending were the chief executives of Barclays, HSBC, Lloyds Banking Group, Royal Bank of Scotland and Santander UK.
The spokesman said: "The banks have put a number of proposals on the table. These proposals will be the subject of further discussions over the Christmas period with the aim of building a sustainable, co-operative relationship between the lenders and the government in 2011 in support of the economic recovery."
He added that the dialogue had been constructive and that the government had "affirmed its desire to see a strong, responsible and internationally competitive financial sector".
Neither the Treasury or Department of Business would elaborate on the proposals the banks, including partly taxpayer-owned RBS and Lloyds, put on the table on both lending targets and the imminent bonus season.
Coalition government members, including Osborne and Deputy Prime Minister Nick Clegg, have urged bankers to show awareness of the rigours people are facing in the wider economy when they award bonuses to their investment banking staff for the year just gone.
One City banking analyst said: "It is something they would never admit, but the lenders probably hope the government will turn down the public heat on remuneration in return for turning on the business credit taps."