By Joan Faus and Graham Keeley

MADRID (Reuters) - Spain's death toll from the new coronavirus outbreak rose by 410 on Sunday, the lowest daily increase in about a month in one of the world's hardest hit countries, prompting cautious optimism from the government that the figures are on a downward path.

The daily increase in deaths was the lowest since March 22. It is far below the highest daily increase - 950 deaths reported on April 2 - in a sign of a slowdown of the spread of the virus after Spain imposed a strict lockdown in mid-March.

The total number of deaths reached 20,453 on Sunday, the Health Ministry said. It is the third-highest toll worldwide after the United States and Italy.

The overall number of coronavirus cases rose to 195,944 from 191,726 on Saturday. Health workers account for 15.6% of those infected, health emergency chief Fernando Simon told a press briefing.

"Data confirms the breaking of the curve, even with an increased number of tests," said health minister Salvador Illa, referring to the evolution of the death toll. "It is still a difficult stage, but we are going in the right direction".

Spain is conducting around 40,000 coronavirus tests daily, one of the highest numbers among European countries, Illa said. Close to a million tests had been conducted as of April 13.

Prime Minister Pedro Sanchez said on Saturday he would ask parliament for a 15-day extension of the country's lockdown until May 9, but said the restrictions would be more flexible.

Parliament will convene next Wednesday, Illa said.

The leftist-coalition government received widespread backing for the previous two extensions of a state of emergency first imposed on March 14, including from the main opposition conservative party PP.

"From the beginning we have said that (Sanchez) would have our support to save lives but not to ruin Spain or to hide his mistakes," PP's leader Pablo Casado said in an interview with La Razon newspaper published on Sunday.

The interview was conducted before Sanchez's announcement on Saturday that he would seek another extension.

Sanchez is scheduled to meet Casado on Monday as the prime minister aims to launch what he has called "agreements for reconstruction", a wide-ranging nationwide pact to address the aftermath of the virus' impact.

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Spain began this week to ease its strict lockdown by opening up some sectors of the economy, but most people are still confined to their homes except for essential outings such as food shopping.

From April 27, children will be allowed to leave their homes for short periods. The details of the policy will be discussed at a cabinet meeting on Tuesday, Illa said.

Separately, the government could consider allowing people to exercise outdoors and letting the elderly go for walks at some future point, emergency chief Simon said.

Sanchez held a meeting on Sunday with Spain's 17 regional leaders, in which he announced a 14 billion euro ($15 billion)financial help package, officials said.

But in a sign of brewing regional tensions in Spain, the Catalan separatist regional head of government Quim Torra said after the meeting that it was his government that should handle the reduction of confinement measures in Catalonia.

That drew a later rebuke from Illa, who reiterated that the government is centralizing decisions under the state of emergency.

(Reporting by Joan Faus and Graham Keeley; Writing by Joan Faus; Editing by Alex Richardson and Hugh Lawson)