A fifth of the 40,000 people surveyed predicted prices will be lower in a year's time, the lowest percentage since autumn 2010, the report said.
The study follows a report from property website Zoopla this week which suggested that house sellers are now less likely to drop their asking prices than they were three months ago and are offering smaller discounts.
Zoopla suggested that buyer demand is remaining strong enough for sellers to hold firm with their prices, despite recent changes to some stamp duty rules which increased costs for buyers, such as the recent removal of a concession for first-time buyers.
Lenders are also tightening their credit criteria, making it harder for borrowers to get a mortgage in the coming months, while more than a million homeowners saw their mortgage payments rise this month due to a string of lenders putting up their rates.
However, among those who believe prices will rise in the Rightmove survey, 35% put their belief down to improvements in the mortgage market, while 14% say record-low interest rates have helped.
Miles Shipside, director at Rightmove, said: "Confidence plays an important role in motivating those who can afford to buy to actually go ahead.
"Many of them are hunting in the same better-heeled locations, which in turn builds greater momentum and price rise expectation in these more affluent areas. Conversely, lower levels of activity in less well-off areas spreads negative sentiment, fuelling falling price confidence."
Among those who believe prices will go down in the next year, nearly two-thirds believe this will happen due to a lack of mortgage availability, high deposits required by lenders or a fear of interest rates rising.
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