Rightmove said asking prices for homes put on sale between May 15 and June 11 were 0.3% higher than a month earlier, down from a 2.1% rise in its May data.
Compared with a year earlier, asking prices are up 9.7%, less than the 10.2% increase recorded for May.
"The exceptional pace of the market is easing a little, as demand gradually normalises and price rises begin to slow, which is very much to be expected given the many record-breaking numbers over the past two years," Rightmove director Tim Bannister said.
Rightmove expects price increases to slow further over this year, to give a 5% annual rise.
Earlier this month Bank of England Deputy Governor Jon Cunliffe said the property market appeared to be slowing, reflecting broader weakness as the public faced the highest consumer price inflation in 40 years.
The BoE has raised rates five times since December to 1.25%, and financial markets expect them to hit 3% by the year's end.
House prices surged in Britain and many other Western countries after the initial onset of the COVID-19 pandemic, as many richer households had spare disposable income and sought more spacious housing for working from home.
Other house price measures from Nationwide Building Society and Halifax have also cooled from recent peaks, though year-on-year price gains remain high.
Rightmove said the number of prospective buyers per home on offer was still more than double its level before the pandemic - and 6% higher than a year ago - but had dropped by 8% over the past month.
In a sign of the bottlenecks affecting many parts of Britain's economy, Rightmove said the legal conveyancing process for buying a home now took an average of five months, 50% longer than in 2019.
(Reporting by David Milliken, editing by Andy Bruce)