The prediction was made by property analyst Hometrack as its latest house price index showed that in May, Bristol became the first city outside the South East to see prices rise at a faster rate than London for more than six years.
Prices in Bristol increased at 14.1% year-on-year, taking it to the top of an index measuring annual growth across the UK's biggest towns and cities.
Prices in London increased by 13.8%, while in Cambridge they rose 13.4%.
Hometrack said it was the first time since January 2010 that a city away from the South East has seen faster price growth than London.
The firm said growth in Bristol is part of a trend in recent months where the recovering economy is leading to stronger price growth in big regional cities such as Bristol, Leeds and Manchester. A stamp duty hike for buy-to-let investors on April 1 also led to stronger housing activity earlier in the year as investors rushed to snap up properties ahead of the deadline.
Richard Donnell, insight director at Hometrack, said: "House price inflation in major cities outside of London and the South East, such as Bristol and Liverpool, has been accelerating but it is now expected to slow towards low single digits in the coming months as demand cools on the back of the EU referendum result."
House price inflation across the UK's major towns and cities accelerated from an annual rate of 10.8% in April to 11.2% in May.
But Hometrack said it expects a "rapid deceleration" in price growth across all cities in the second half of the year as potential buyers wait to see how the economy will be affected, with property sales numbers also being hit.
The immediate impact will be felt in London, where the affordability of a home is already particularly stretched, Hometrack predicted.
It said modest single digit price falls "appear likely" in higher value property markets.
In Aberdeen, where a downturn in oil prices is impacting on the economy and demand for housing, prices are already falling by 9.6% annually, the report said.
But Hometrack said it does not expect to see widespread house price falls as the biggest immediate impact will be on housing market activity rather than prices.
Price falls would require sellers to be under pressure from higher mortgage rates and/or rising unemployment, it said.
A tight supply of homes is also likely to help to support prices, and new levels of housing from builders are also likely to be scaled back, the report predicted.
The impact for housing markets up and down the country will be driven by the local economies and jobs markets in these areas, it said.
Mr Donnell said: "Standing back from the immediate turmoil in financial markets, the reality is that the fundamentals of the housing market remain unchanged with record low mortgage rates and a wide imbalance between supply and demand.
"The UK doesn't have a problem with housing demand, the more important question is how many buyers and sellers feel confident to participate in the market in the near term.
"Market sentiment can change quickly and the sooner a clear picture emerges over the likely impact on the economy and the outlook for jobs and mortgage rates the sooner transaction volumes should stabilise and more buyers return to the market."
Here are the average May house prices in the 20 towns and cities monitored by Hometrack and the year-on-year price growth:
- Bristol, £250,900, 14.1%
- London, £472,100, 13.8%
- Cambridge, £410,200, 13.4%
- Southampton, £212,600, 10.3%
- Bournemouth, £264,600, 9.1%
- Portsmouth, £213,300, 9.1%
- Birmingham, £141,800, 8.2%
- Manchester, £144,300, 7.9%
- Oxford, £401,600, 7%
- Leeds, £149,900, 6.9%
- Liverpool, £112,800, 6.5%
- Cardiff, £188,200, 6.4%
- Nottingham, £134,500, 5.8%
- Leicester, £149,000, 5.8%
- Sheffield, £127,500, 4.5%
- Newcastle, £125,600, 3.6%
- Edinburgh, £199,300, 3.4%
- Glasgow, £111,700, 3.4%
- Belfast, £122,400, 3.2%
- Aberdeen, £182,100, minus 9.6%