At £743 on average, monthly rents in August were just £1 less than an all-time high recorded in October 2012, according to LSL Property Services, which owns chains Your Move and Reeds Rains.
The pace of rent increases stepped up to 0.7% month-on-month in August, following a 0.2% monthly uplift in July.
Rents are 1.3% higher across England and Wales than a year ago, showing an increase which is still less than half the rate of consumer price index (CPI) inflation at 2.7%.
But at £1,126 typically, rents in London have risen at a much faster rate than inflation and are up by 4.8% year-on-year.
Official figures released yesterday showed that house prices in London are also up by nearly 10% year-on-year, indicating the strength of demand for homes in the English capital.
Wales saw the second biggest annual increase in rents, with a 2.3% uplift taking average rents to £561. The South East recorded the strongest month-on-month growth, with a 2% rise pushing monthly rents to £762.
The North East saw the biggest month-on-month drop in rents, with a 0.8% fall taking average rents to £523.
LSL's findings are based on rents achieved on around 20,000 properties and its records go back to January 2008.
Across the country, rental inflation had been cooling off for much of this year following the launch of Government schemes to give people with low deposits a helping hand onto the property ladder.
First-time buyer numbers have reached their highest levels in more than five years following initiatives such as Funding for Lending and Help to Buy, which have widened access to mortgages and allowed some people who were previously trapped in renting to break free.
But David Newnes, director of LSL Property Services, said that weak income growth, which has an impact on households' ability to borrow, and a lack of housing supply means that the private rental sector is continuing to see strong demand from new tenants.
He said the upward pressure on rents is also coming from an uplift in student renters returning to the market as the new academic year begins.
Mr Newnes said: "Better availability of finance has allowed some households to leave the rental market. And rents certainly felt the short-term impact of that.
"But releasing a blast of pent-up pressure to buy a home is unlikely to change the long-term trend in renting.
"Although Government schemes are helping, buying a first home is still extremely hard on the back of low salary growth."
Tenant finances also improved in August, with 7.8% of rent late or unpaid during the month, down from 8.1% in July.
Mr Newnes put the improvement in part down to the softening of rental inflation seen earlier this
year which gave tenants some "relief".
Housing Minister Mark Prisk said: "Official Valuation Office Agency figures show that median rents were unchanged nationally over the past year.
"However, this Government is determined to kickstart a bigger, better private rented sector that gives tenants more choice.
"That's why we're getting Britain building more rented homes, with a £1 billion investment to help developers build to rent and a further £10 billion in loan guarantees to help grow the number of rented and affordable homes.
"For those looking to make the move from rent to home ownership, there is a range of Government-backed schemes on offer, which have already helped nearly 26,000 hard-working people achieve their dreams of buying a home.
"We've also delivered 150,000 affordable homes in England over the last three years, meaning we're heading towards the highest rate of affordable housebuilding for two decades."