The Bank of England is considering a request by Britain's big five banks to reform the way it provides them with emergency funding.
Mr King is sympathetic to requests for more help from the banks
Governor Mervyn King has told bank bosses he is sympathetic to appeals for more help, the BBC has learned.
The banks want to be able to use a wider range of collateral, including mortgages, for Bank of England loans.
Mr King had been reluctant to allow that so he would not be seen as bailing out banks that had behaved foolishly.
The Bank of England is said to be considering whether it can provide support similar to that of the US Federal Reserve.
Chief executives from HSBC, Royal Bank of Scotland, Barclays, Lloyds TSB and HBOS held a meeting with Mr King on Thursday.
It is understood the governor was alarmed by the recent fall in the HBOS share price on the back of erroneous rumours that the leading mortgage bank was in financial difficulties.
The BBC's business editor Robert Peston said the bank bosses believe they would be less vulnerable to damaging speculation if the Bank of England said it was prepared to make good any hole in their finances stemming from the current crisis in banking markets.
"Although the Governor of the Bank, Mervyn King, asked them not to divulge what they discussed, I have learned he signalled - for the first time - that he was sympathetic to their request that in an emergency they should be able to swap a wider range of assets, including their mortgages, for loans from the Bank of England," the BBC's business editor said.
Before the meeting with bank chiefs, the Bank of England injected an extra £5bn into the money markets in an attempt to ease concerns about the liquidity of the banking system.
The loans were added to the normal weekly funds offered to commercial banks, making £11bn now available.
Three times over-subscribed
The Bank said an extra £5bn a week would be released at least until its next monthly meeting on interest rates on 9 April.
This week's additional funding was three times oversubscribed and both the European Central Bank and the US Federal Reserve have made much more money available to their banks with wider collateral accepted.
HBOS shares were among the top gainers on the London Stock Exchange on Thursday, up 6.2% recovering most of their falls from the previous day.
On Wednesday, the Financial Services Authority (FSA) said it would "not tolerate" traders starting "false" rumours about firms to make cash from dealing in their shares.