It was the second consecutive quarter that lettings demand has risen at a pace above the long run average.
Tenant demand increased across all regions, but was strongest in London and the East of England.
Continued difficulty in securing mortgage finance, worries over a double dip in housing and large deposits required by lenders are leading to higher numbers seeking to rent rather than buy.
Rents increased for the second consecutive quarter, with 27%more surveyors reporting a rise in rents than a fall.
A year ago 29% more surveyors reported falling not rising rents.
Difficulty in securing buy-to-let mortgages is contributing to the lack of supply.
New supply of property to the market remains low and has now fallen for four consecutive quarters, although at a slightly slower pace.
In the run up to July the net balance of surveyors reporting a fall in landlord instructions was -6, in comparison to a net balance of -12 in the previous quarter.
But just 4.1% of existing landlords say they intended to sell their properties at the end of a tenancy agreement.
One third more surveyors expect rents to increase over the next quarter rather than fall.
Rents for houses are expected to marginally outperform flats, with the net balances for this forward looking indicator moving to +34 and +31 respectively.
James Scott-Lee, spokesman for RICS, says: “Supply of lettings property continued to fall in the three months to July although at the slowest pace in a year which amid rising tenant demand has helped propel rents higher for the second consecutive quarter. Existing landlords keen to expand their portfolio may still be struggling to access the necessary finance despite improved market conditions.
“However, there is a possibility the lettings market could face a modest increase in supply in the coming months. The latest RICS Housing Market Survey shows a lack of funding has stifled demand from buyers which may cause some moderation to rents as more opt to let than sell.”