European and US stocks rallied Tuesday as data boosted hopes of a quick economic recovery and investors continued to shrug off increased coronavirus cases in many American states.

Private sector economic activity in the eurozone and Britain remained in contraction in June, but the rates of decline slowed considerably compared with the worst of the coronavirus lockdown, IHS Markit said Tuesday.

Jessica Hinds, European economist at independent analysts Capital Economics, said the survey "suggests that while the GDP outturn will undoubtedly be dreadful, it will not be as catastrophically bad as we had feared."

In the US, new home sales jumped 16.6 percent in May above April to a seasonally adjusted annualized rate of 676,000, according to the Census Bureau.

"Pent up demand ahead of the virus outbreak and households' shifting preference for more space as well as lower mortgage rates are likely boosting sales," said Rubeela Farooqi of High Frequency Economics.

The main European markets gained more than one percent, while Wall Street stocks also powered higher, lifting the Nasdaq to its second straight record.

Investors largely overlooked big increases in coronavirus counts in several US states, including in Texas, Oklahoma and South Carolina.

Cases are also increasing in Florida, where Governor Ron DeSantis said bars and restaurants that violate social distancing rules risk losing their liquor licenses, according to local media.

The coming two weeks will be "critical to our ability to address those surgings," said Anthony Fauci, the nation's top infectious disease expert.

Last week, Apple announced that it was re-closing 11 stores in four states that have seen jumps in COVID-19 cases.

Art Hogan, chief market strategist at National Securities, said the market is "fragile," with investors watching if more companies follow Apple and close down again.

"If in fact we start to hear more steps backward, the market has not priced that in," he said.

Markets also shrugged off comments from White House advisor Peter Navarro casting doubt on the US-China trade agreement.

Navarro, in the course of an interview with Fox News, said the deal was "over" and blamed the coronavirus outbreak in the United States on China, amplifying a grievance that Trump himself has championed.

The remarks rattled global markets and prompted a quick selloff.

New York City, once the epicentre of the US outbreak, has moved into a new phase of reopening, as countries around the world slowly emerge from strict lockdown measures
New York City, once the epicentre of the US outbreak, has moved into a new phase of reopening, as countries around the world slowly emerge from strict lockdown measures AFP / Johannes EISELE

But Trump on Twitter said subsequently that the agreement is "fully intact," reviving the market.

New York - Dow: UP 0.5 percent at 26,156.10 (close)

New York - S&P 500: UP 0.4 percent at 3,131.29 (close)

New York - Nasdaq: UP 0.7 percent at 10,131.37 (close)

London - FTSE 100: UP 1.2 percent at 6,320.12 (close)

Frankfurt - DAX 30: UP 2.1 percent at 12,523.76 (close)

Paris - CAC 40: UP 1.4 percent at 5,017.68 (close)

EURO STOXX 50: UP 1.8 percent at 3,298.83 (close)

Tokyo - Nikkei 225: UP 0.5 percent at 22,549.05 (close)

Hong Kong - Hang Seng: UP 1.6 percent at 24,907.34 (close)

Shanghai - Composite: UP 0.2 percent at 2,970.62 (close)

West Texas Intermediate: DOWN 0.9 percent at $40.37 per barrel

Brent North Sea crude: DOWN 1.0 percent at $42.63 per barrel

Euro/dollar: UP at $1.1308 from $1.1261 at 2100 GMT

Dollar/yen: DOWN at 106.53 yen from 106.91 yen

Pound/dollar: UP at $1.2520 from $1.2469