Halifax said there are tentative signs that a better balance between demand and supply may be emerging which, if sustained, would help to dampen the pace of house price growth.
The number of new buyer enquiries fell marginally in July; the first decline since January 2013 whilst new sales instructions increased for the second successive month.
Martin Ellis, Halifax housing economist, said:"Housing demand is supported by continuing economic recovery, growth in employment, improving consumer confidence and low mortgage rates.
“Nonetheless, earnings growth that remains below consumer price inflation, and the prospect of an interest rate rise at some point over the coming months, are likely to curb demand.
“There are some signs of an improvement in housing supply, both in terms of more second-hand properties coming onto the market and increased numbers of new homes.
“These trends, if sustained, should help to improve the balance between supply and demand, contributing to an easing in the pace of house price growth."
The Halifax also found that home sales have remained broadly stable so far this year - at between 101,000 and 104,000 a month - with the exception of February (109,500).
Jonathan Samuels, chief executive of Dragonfly Property Finance, thinks a period of housing market stability is just what the sector needs.
He said: "Less momentum and more moderation in house price growth is what the market needs. And for the time being, that's what it's getting.
"Increased supply on a consistent basis will further temper the rate of price growth and offer hope to prospective buyers.
"Growing consumer confidence, a strengthening jobs market and economy, and historically low mortgage rates have kept demand ticking over.
"But at the same time, a number of factors have combined to slow the market down slightly, from weak wage growth and the growing anticipation of higher interest rates to tougher mortgage criteria.
"As ever, there is no single narrative for the UK property market. In some areas of the country, prices are looking dangerously inflated, while in others the market is still relatively weak.
"Prospective buyers should remain on their guard and factor rate rises into the buying equation."