The attack has also hit 150 countries around the world since Friday.
"It is important to note that the vast majority of NHS organisations report that they are running contemporary IT systems, which are commissioned depending on local need", the NHS said in a statement.
Mr Putin said the incident was "worrisome" but had done "no significant damage" in Russian Federation and called for urgent global talks on countering the hackers.
Russia's largest bank Sberbank said its systems "detected in time attempts to penetrate bank infrastructure".
But experts and government officials alike warn against giving in to the hackers' demands. In a scathing statement over the weekend, he said an equivalent scenario in conventional weapons would be the US military having some of its Tomahawk missiles stolen.
Microsoft Corporation (NASDAQ:MSFT) said the attack should be treated around the world as a "wake-up call", saying software vulnerabilities hoarded by governments have caused "widespread damage".
But global fears eased on Monday as the number of incidents reported levelled off.
Friday's ransomware attack hit over 125,000 computer systems, with 48 National Health Service (NHS) trusts having to cancel appointment and operations. On Twitter, he urged users to immediately install a security patch for older versions of Microsoft's Windows, including Windows XP.
U.S. package delivery giant FedEx, European auto factories, Spanish telecoms giant Telefonica, Britain's health service and Germany's Deutsche Bahn rail network were among those hit.
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In the United Kingdom more than 50 hospitals, doctor surgeries and pharmacies were targeted, Germany's rail network Deutsche Bahn was attacked, and Russia's Interior Ministry was infected with the malware.
"We are obviously continuing our investigations but it does seem the figure has gone from one to three", he said.
"We haven't fully dodged this bullet at all until we're patched against the vulnerability itself", Kalember said.
The virus is then able to encrypt files and block user access to them, displaying a pop-up window onscreen telling users they have been blocked and demanding payment.
Pictures posted on social media showed screens of NHS computers with images demanding payment of 300 U.S. dollars worth of the online currency Bitcoin, threatening to delete files within seven days. They can cash out easily thanks to the use of the Bitcoin virtual currency, which is hard to trace.
Speaking to ITV's "Peston on Sunday", Europol director Rob Wainwright said the attack was indiscriminate across the private and public sectors.
Security firm Digital Shadows said on Sunday that transactions totalling $32,000 had taken place through Bitcoin addresses used by the ransomware.
While the scale of the attack shows Microsoft needs to strengthen its own capabilities, "there is simply no way for customers to protect themselves against threats unless they update their system", Smith said in his blog post.