UK house price growth was "sluggish" in May, according to the latest survey from the Nationwide Building Society.
Affordability is still a problem for first-time buyers
Prices rose by only 0.2% this past month, pulling back the annual rate of house price inflation from 4.8% in April to 4.7% in May.
Prices in the three months to May rose by 1.6% compared with the previous quarter, down from the 1.7% rate seen in the three months to April.
The average house price in the UK is now £164,632, up by £7,500 in the year.
Further evidence of a possible slowdown came in figures from the Bank of England.
They showed that the number of mortgages approved for house purchase dropped in April to 106,000.
That was down by 8,000 from the previous month and was significantly lower than the average of 115,000 over the preceding six months.
The Royal Institution of Chartered Surveyors (RICS) said that a survey of its members also suggested that the market was not returning to a boom.
"A modest slowdown in the housing market is backed up by RICS estate agents who have reported that new enquiries from would-be buyers rose in April at the slowest pace in almost a year" a spokesman said.
The Nationwide, the UK's largest building society, said there were signs that the market may now be starting to cool.
House prices have generally been accelerating since the start of the year.
But the society's chief economist Fionnuala Earley said: "The numbers of house purchase approvals remain well above their long term average, but they have begun to moderate since the start of the year. We expect them to cool further during the summer to around their long-term trend. "
She also pointed out that enquiries from new buyers at estate agents have also been falling since the end of last year.
The Nationwide added that further demand for houses would now depend on whether the Bank of England increases interest rates this year.
"In our view, the decision is still very close," said Nationwide group economist Fionnuala Earley.