The former Russian spy, 66, and his 33-year-old daughter were found slumped on a bench in the cathedral city eight days ago, suffering the effects of nerve agent. They are still in a critical condition.
Nick Bailey, a police officer who was among the first on the scene, was also affected but is said to be in a stable condition, conscious and talking.
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The Times newspaper reported that senior government sources have said police and security services have established enough evidence to link Moscow to the attempted killings.
Mrs May is chairing a meeting of the National Security Council later, which will include senior ministers, intelligence and military chiefs.
An announcement could come after that, along with possible repercussions, which could include expelling Russian diplomats and spies and new financial sanctions.
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But some say Russia is unlikely to be behind the attempted murders.
Nikolai Tsiplakov told The Associated Press that he had lived in the same building as Mr Skripal’s family in Moscow before the retired military intelligence officer was jailed for spying for Britain.
Mr Tsiplakov said there was no proof of Russian involvement and added You need to understand, he’s a former spy and he’s a traitor.
If they wanted him dead, they would have killed him in Russia a long time ago.
Mr Skripal was released from a Russian jail in 2010 and brought to Britain as part of a spy swap.
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The news comes after hundreds of people were told to wash their clothes and other items in case they had also come into contact with the nerve agent, which has not yet been publicly identified.
The advice was given after traces of the substance were found at The Mill pub and Zizzi restaurant, with up to 500 people thought to have been in the venues between Sunday lunchtime and Monday night.
People were told to machine wash clothing worn on the day, and double-bag items which would normally be dry cleaned to await further advice.
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Officials insisted that the risk to the public was still low, however.
Also, Jamie Knight, 30, from Salisbury, will appear before magistrates in Swindon on Monday accused of breaching one of the cordons on Friday evening.
News Source SkyNews