Across England and Wales, the typical monthly private sector rent reached £846, the highest amount since the study from lettings agent Your Move started in 2008.
July's figure is 5.2% higher than a year earlier and a £36 jump on the average rent of £810 recorded in June 2016, suggesting the decision to leave the EU has had no effect so far on pushing rents downwards, the report said.
The increase in rents also comes despite a wave of rental properties being snapped up by landlords earlier this year, ahead of a stamp duty hike for buy-to-let investors which was imposed from April 1. The extra properties bought up could have helped to widen the choice of rental homes for some tenants.
Adrian Gill, director of Your Move, said: "The UK's vote to leave the European Union has not caused any immediate change in the rental market, although we must wait for longer term trends to develop."
Growth in rents in the South East of England was particularly strong, with the typical monthly rent now standing at £924 - 14.9% higher than a year ago.
Mr Gill said: "The South East was home to the biggest leap in rents, with many Londoners moving further afield in an attempt to escape high rents in the capital."
London has the highest average monthly rents, at £1,273. In Wales, the average rent now stands at £618.
Tenants' finances worsened in July, with 9% of tenancies having arrears of a day or more, up from 8.4% in June.
But the proportion of tenants in arrears remains well below a high of 14.6% recorded in February 2010.
The report, which is compiled in partnership with the Centre for Economics and Business Research (Cebr) is based on analysis of around 20,000 properties across England and Wales.