If a solution is found, up to a million British Muslims would be able to open a bank account in this country for the first time.
"A lot of them keep their money at home," said Muazzam Ali, chairman of the Institute of Islamic Banking and Insurance. "If Islamic banking was allowed, we believe a lot of Muslim people would come forward."
Around 18 banks in Britain offer non-interest-based mortgages and investment schemes geared to Muslims.
However, according to Bank of England rules, a bank can only be licensed as such if it takes deposits and pays interest. But now that institutions such as supermarkets have entered financial services, Muslim groups are turning to the regulator in the hope of finding a solution.
The Institute of Islamic Banking, the Islamic Foundation and the Muslim Council of Britain will meet Mr Davies on 8 September. "This is a very important step forward. They are listening," said Richard Thomas, deputy chief executive of the Islamic Banking Unit of the United Bank of Kuwait.
Regulators have been wary of Islamic banking practices ever since the collapse of the Bank of Credit and Commerce International (BCCI) in the early 1990s. BCCI did not practise Islamic banking but many of its clients were British Muslims. The only UK Islamic bank, Alabarak Bank, had its licence revoked in 1993 by the Bank of England.
All other Islamic banks are based in Muslim countries, with Bahrain as their financial centre. There is an estimated $100bn (pounds 62bn) invested with Islamic banks.
Islamic banks operate like a large mutual fund, investing in equities markets apart from companies with interests in banks, casinos and alcohol. The basic religious concept is that the rich shouldn't get richer simply by having money, but also that funds should be put to productive use through corporate investment.
"It's a cross between ethical and merchant banking," said Mr Thomas.
The Bank of Kuwait is the only bank in the UK to offer "halal mortgages", by which the bank purchases the house and the occupants pay the loan back in the form of rent.