Putin and Russian Federation have vehemently denied having anything to do with the poisoning cases.
Theresa May previously announced the pair were Russian military intelligence officials, there was "hard evidence" against them and "this was not a rogue operation. nearly certainly" approved at a "senior level of the Russian state".
The British government recently said Scotland Yard detectives had identified Mr Petrov and Ruslan Boshirov as suspects, who they said were from Russia's foreign military intelligence agency, the GRU.
Two days later, they sprayed nerve agent Novichok on the front door of Skripal's home in the Wiltshire city of Salisbury, before travelling home to Russian Federation later that day, the police said.
The duo had entered Britain on genuine passports, prosecutors said, while British Prime Minister Theresa May described them as military intelligence officers nearly certainly acting on orders from high up in the Russian state.
Putin urged the men to speak to journalists.
"We know who they are, we have found them", Mr Putin told an economic forum in the presence of Japanese Prime Minister Shinzo Abe and Chinese President Xi Jinping in the far eastern city of Vladivostok.
There's nothing special and criminal about it, I assure you.
But Mr Putin said they were civilians and would tell their story soon. The Skripals were hospitalized for months but recovered.
"The GRU is a highly disciplined organization with a well-established chain of command, so this was not a rogue operation, it was nearly certainly also approved outside the GRU at a senior level of the Russian state", she said in a statement to the House of Commons last week. "They're civilians, of course".
"We have repeatedly asked Russian Federation to account for what happened in Salisbury in March, and they have replied with obfuscation and lies", Prime Minister Theresa May's spokesman told reporters. "I can see nothing to suggest that has changed". "They will come somewhere, to you, the mass media".
The pair flew back to Moscow from Heathrow later that night.
The suspects fled to Russian Federation, who won't extradite them, but they would be arrested if they entered any European Union country.
Police said Ms Sturgess and Mr Rowley were later exposed to Novichok after handling a contaminated container, labelled as Nina Ricci Premier Jour perfume.
"But the question of course is who will we see because don't forget that, in the United Kingdom, the suspicion is that the two names that were given are in fact aliases", our correspondent added. Russian news agency Fontanka has found traffic tickets and a Moscow apartment registered to Boshirov, but none of the neighbors know him.
While one of the two accused, Boshirov, looked like a photograph released by Britain, the other, Petrov, looked nothing like the other photograph and had not been in Britain at the time of the attempted poisoning incident, she added.
His hint that the men could soon break their cover recalls memories of the assassination of former KGB agent Alexander Litvinenko on British soil, when suspects Andrey Lugovoy and Dmitry Kovtun publicly rejected the allegations from Russian Federation.
A British inquiry concluded that Litvinenko had been killed at the behest of the Russian state, probably with the president's knowledge. Britain spent years trying in vain to prosecute the prime suspects, Andrei Lugovoi and Dmitry Kovtun.
The Russian government rejected the accusations and was quick to throw its support behind the men.
Associated Press writer Jim Heintz in Moscow and Jill Lawless in London contributed.