SINGAPORE/LONDON, Dec 5 (Reuters) - The euro and pound hit over five-month highs against the dollar in early trade on Monday as improved sentiment caused by Chinese cities easing some COVID-related restrictions helped investors justify continuing their shift away from the greenback.

Official messaging about how dangerous the virus is has softened and financial hub Shanghai and Urumqi in the far western Xinjiang region were among the cities that announced an easing of coronavirus curbs over the weekend following recent, unprecedented protests against the government's uncompromising "dynamic zero-COVID" strategy.

This boosted China's yuan, and the dollar fell below 7.0 yuan in offshore trade for the first time since mid September, and lost 1.36% on the onshore yuan to as low as 6.9473 on Monday, its weakest since Sept. 13.

The consequent optimism also helped European currencies to rally against the safe haven greenback. The euro hit $1.0585 in Asian hours, its highest since June 28, and sterling hit $1.2345, its highest since June 17.

Both currencies gave back those gains by London trading and the European common currency was flat at $1.0546 and the pound down 0.1% at $1.2268.

"It may seem like they are baby steps but nonetheless quite a strong sign of China taking calibrated steps in the direction of reopening," said Christopher Wong, a currency strategist at OCBC in Singapore.

China is soon set to announce a nationwide easing of testing requirements as well as allowing positive cases and close contacts to isolate at home under certain conditions, people familiar with the matter told Reuters last week.

Much of the dollar's recent weakness has also been driven by expectations the Federal Reserve will be less aggressive when it comes to further monetary tightening having raised rates by 75 basis points at each of its last four meetings.

The dollar index, which measures the currency against six major peers including the yen and euro, hit 104.1 in early trading Monday, its lowest since June 28.

The index fell 1.4% last week, and 5% in November, its worst month since 2010. Its aggregate positioning against G10 currencies is now neutral, and at the lowest levels since August 2021, according to ING calculations based on CFTC data.

ING feel however, that dollar softening may have run its course for now given the possibility the Fed actually maintains its hawkish narrative for longer, that relaxing China's COVID restrictions could prove complicated, and that oil and gas prices could rise again.

The Japanese yen, which has been one of the greatest beneficiaries of the weaker dollar in the past month was on the back foot on Monday, with the dollar up 0.7% to 135.29 yen.

Moves in other majors were largely in line with the bigger picture -- the China-sensitive Aussie dollar was up a touch at $0.68065, and the dollar slid 0.25% on the Swiss franc to 0.9342, just above Friday's near eight-month low of 0.9326.

(Reporting by Ankur Banerjee in Singapore and Alun John in London; Editing by Stephen Coates, Simon Cameron-Moore and Crispian Balmer)