However, the net balance for new instructions rose to the highest level for three years impacting on sentiment for future price rises.
Ten per cent more chartered surveyors reported a rise than a fall in house prices down from 22% in May.
Surveyors are still reporting house price rises in most parts of the country but the increase in supply is pushing many of the regional net balances towards negative territory.
The most notable exceptions to this trend are London and Scotland.
Buyer interest fell for only the second time since October 2008 reflecting in part heightened uncertainty over the near term outlook for the economy.
Five per cent more chartered surveyors reported a fall than a rise in new buyer enquires down from a positive 8% in May. This contrasts with a sustained rise in supply.
The net balance of surveyors reporting rises in new instructions has remained in positive territory for 13 successive months. 27 per cent more chartered surveyors reported a rise than fall in new instructions, up from 22% in May and the highest reading since May 2007 when the net balance was 44%.
This latest increase in instructions is partly a response to the recent decision to abolish HIPs in England and Wales, according to RICS.
Commenting on the data, Jeremy Leaf, RICS spokesperson, said: “A shortage of stock has been one factor holding back transaction activity in the housing market but the abolition of HIPS is helping to belatedly address this issue.
“This is likely to be reflected in higher sales numbers over the coming months. However, with supply of property now beginning to outstrip demand there is a risk of some modest slippage in prices during the second half of the year.”