The Nationwide said the number of people joining the property ladder rose 5 per cent in the first three months of the year - the highest increase since summer 1997.
Last year about 35 per cent fewer first-time buyers entered the market compared with the peak reached in the 1980s.
In a separate report, the UK's leading banks reported mortgage lending in April was 20 per cent up on a year ago, for the second month in a row.
In its May housing index, Nationwide said annual inflation was running at 7.4 per cent, compared with 7.1 per cent in April. Prices rose 1.2 per cent between April and May, taking the average price of a home to pounds 69,933. The previous monthly rise was just 0.1 per cent.
Philip Williamson, Nationwide marketing and commercial director, said: "Central to the improved outlook for the housing market is renewed interest from first-time buyers.
"This is likely to have been driven by higher real-income growth, low mortgage rates and continued jobs growth. With the wider economy looking set for a soft landing and consumer confidence surging, housing market activity should improve further this year."
The figures back up official data from the Land Registry for the first quarter, which showed prices of new flats and maisonettes - the typical first buyer's home - rose 22 per cent.
Annual house price inflation of 7.4 per cent is ahead of both the Nationwide's own forecast of 4.5 per cent and the Halifax's figure of 3.7 per cent, which may reflect the north-south split between the two lenders.
Mr Williamson warned that southern England had seen stronger gains than anywhere else, as other regions had not been able to offset manufacturing job losses with new jobs in other sectors. "This pattern is likely to persist for much of the year, though London price rises may temper as buyers look to surrounding areas in search of value," he said.
The British Bankers' Association said mortgage lending by the big banks surged 20 per cent last month compared with a year ago.
Gross mortgage lending by the main British banks totalled pounds 5.18bn compared with March's total of pounds 4.97bn. The number of new home loans, at 63,151, also rose at an annual rate of 20 per cent, it said.
The association said comparisons were confused by building societies converting into banks over the past year, but said the underlying trend showed 20 per cent growth.
"The major banks' mortgage business maintained a higher level of activity, in excess of the usual seasonal upturn at this time of year," said Tim Sweeny, the association's director general.
It also reported that new lending on credit cards rose by pounds 487m in April.