There was also upbeat news from property website Rightmove which said there were growing signs that the past year's price falls have bottomed out, and from the Council of Mortgage Lenders, which said gross lending hit a six-month high in June.
However, mortgage lending is recovering from a very low base, and economists said the weakness of business lending and the biggest monthly contraction in the money supply since June 2005 did not bode well for a strong recovery.
"Altogether there are tentatively encouraging signs but it's difficult to be too optimistic," said Philip Shaw, chief economist at Investec.
The Bank of England's Trends in Lending report showed the country's top six banking groups approved 51,000 mortgages for house purchase last month, up from 45,000 in May. Gross mortgage lending picked up to £9.2 billion from £8.6 billion.
Net lending to businesses fell by £3.4 billion in May after a £6 billion fall in April. That pushed the annual rate of growth down to 0.1% in May from 1.1% in April.
A pick-up in lending is vital to recovery prospects after the economy contracted at its sharpest rate in the first quarter since 1979.
The government has been urging major banking groups to pass on credit to the rest of the economy after a series of costly and unprecedented initiatives to shore up the financial system.
The Bank of England has cut interest rates to a record low and embarked on an £125 billion asset-buying program in a bid to get credit flowing more freely again.
The Bank survey showed that while lending to businesses remained weak, credit flows could improve over the next few months.
"A stabilisation in the economic outlook, as well as slightly more plentiful and cheaper funding, was expected to help them make credit more available over the next three months," the report said.
The Council of Mortgage Lenders said gross mortgage lending rose to £12.3 billion in June, its highest level this year but still almost 50% lower than a year ago.
But it cautioned that the pace of improvement was unlikely to last.
"The pick-up in June's lending largely reflects seasonal factors and these may well support lending at moderately higher levels over the summer," said CML economist Paul Samter.
"But the combined effects of the restricted nature of mortgage funding, reduced number of active lenders, weak labour market and limited consumer demand are likely to hold back any significant and underlying improvement."
Separate data from the Bank showed the M4 measure of broad money supply contracted 0.2% on the month in June, the weakest reading since June 2005.
On the year, broad money supply growth eased to 14.2% in June from 16.7% in May, the weakest since September 2008.
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