House sellers' asking prices rose by 8.9% in the year to May, meaning the average property is on sale at a new record high of £272,003.
The annual rate crept closer to the 10.4% seen in October 2007 while a month-on-month increase of 3.6%, or £9,409, was the highest ever seen for the time of year.
London led the way with a 16.3% year-on-year increase, compared with a more modest 4.9% in the rest of the country - suggesting that fears of a bubble were not borne out outside the capital, Rightmove said.
The average asking price in the city was up by nearly £80,000 so far in 2014, or £4,405 a week, compared with £1,521 a week for the rest of the country.
Ten out of 32 boroughs in London saw annual rises of more than 20%, with a 43% increase in Tower
Hamlets driven by cash buyers and investors in Canary Wharf and neighbouring areas.
Rightmove said that across the country demand remained strong but the supply of new properties to the market was still unable to keep pace.
An increase in new sellers earlier in the year was reversed in May as the number of homes being newly put up for sale fell 1% compared with April. It was thought to be the result of the timing of bank holidays affecting activity.
The figures come after fears of an overheating housing market failed to persuade the Bank of England to raise interest rates, with governor Mark Carney dampening expectations of an early hike.
Policy makers have said they will use other tools first, if needed, to cool the property market, before turning to this "last line of defence".
Earlier this month, the Bank's deputy governor Sir Jon Cunliffe warned that surging house prices could pose the biggest danger to the country's financial stability and it was the brightest of the "blinking warning lights" of risk facing the UK.
Meanwhile, the Organisation for Economic Co-operation and Development has also recently sounded a warning that action may be needed to cool the market, calling for curbs on the Government's Help to Buy scheme.