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.
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.
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.