France, Italy and Spain have registered the lowest daily death tolls for weeks as they prepare to ease restrictions.
France reported 135 deaths; Spain's 164 were the lowest since mid-March; and Italy's 174 was a two-month low.
Meanwhile, a French doctor has claimed new tests on samples from patients show the virus was present in the country last year - weeks before the first officially recorded case.
In Russia the virus appears to be advancing, with 10,000 new infections.
But Russia's mortality rate remains low relative to other countries. On Sunday it recorded 58 deaths, taking its total to 1,280.
The UK recorded 315 new deaths and has the third-highest number of deaths behind the US and Italy. However, British officials say the outbreak has peaked and the number of new hospital admissions is declining.
Globally the pandemic has seen nearly 3.5 million people infected and nearly 250,000 people have died.
Was the virus in France last year?
The number of new recorded deaths is the lowest since late March, when only deaths in hospitals were being recorded. The new figures include deaths in care homes and have been declining for several days.
Meanwhile, an intensive care chief in the Paris region has told local media that the virus was present in France on 27 December - a month before the first cases were confirmed.
Yves Cohen told broadcaster BFMTV that his team had revisited negative tests for flu and other coronaviruses on 24 patients who had been in hospital with respiratory symptoms in December and January.
"Of the 24 patients, we had one positive result for Covid-19 on 27 December when he was in hospital with us," he said, adding that the test had been repeated several times to confirm the result.
Dr Cohen said he had reported the case to the regional health authorities and called for other negative tests from the same period to be re-examined.
France is planning to lift its lockdown on 11 May, when children will return to school in phases, some businesses will reopen and people will be able to travel within 100km (60 miles) of their homes without the need for a document giving a reason for their movement.
However, on Sunday Health Minister Olivier Véran said this would depend on further falls in the number of new infections, particularly in the worst-affected areas like the Paris region and northeast France.
France also clarified that a rule requiring anyone entering the country to go into isolation for two weeks did not apply to people arriving from EU countries, the Schengen area or the UK.
What's happening in Spain?
Spain's death toll of 164 was the lowest for a month-and-a-half and was more than a hundred down on Saturday's figure.
On Saturday adults were able to exercise outdoors for the first time in seven weeks. The lockdown was eased for children under 14 a week ago.
Prime Minister Pedro Sánchez said Spain was reaping the rewards of the sacrifices made during the lockdown, one of Europe's strictest.
On Monday masks will become compulsory on public transport and some small businesses such as hairdressers will open for individual customer appointments.
What about in Italy?
Italy registered the lowest daily number of coronavirus-related deaths since just after its lockdown started two months ago. Both the numbers of new infections and patients in intensive care have continued to fall.
The country is relaxing its lockdown from Monday, when Italians will be able to exercise as long as they respect rules on maintaining physical distance. They will also be able to visit relatives - but not friends - within their region.
However schools, cinemas and most shops will stay shut. Bars and restaurants are due to start allowing customers to sit at tables in June.
Italy's death toll stands at 28,884 - second only to the US where more than 68,000 people have died.
The UK's death toll is 28,446 while in Spain and France 25,100 people and 24,729 people respectively have died.
The UK government is expected to announce the next steps in its response to the pandemic next Sunday.