The gardens have been closed off to the public for two weeks since Dawn Sturgess and her partner, Charlie Rowley, fell ill following exposure to a Novichok nerve agent in late June, four months after the poisoning of former Russian spy Sergei Skripal and his daughter Yulia Skripal.
The source added that the suspects' identities were confirmed by cross-checking security footage with records of people entering the country around that time.
They said investigators were "sure" the suspects were Russian.
Wiltshire Police said the search of Queen Elizabeth Gardens was taking place to assist the wider investigation and with a view to safely returning the gardens to public use.
Mother-of-three Ms Sturgess died in hospital a week after falling ill.
Last week counter-terrorism detectives revealed they had found a small bottle containing Novichok at Mr Rowley's Amesbury home.
Sky's Crime Correspondent Martin Brunt said: "This would chime with the government's long-held belief that the Russian state was behind the attack on the Skripals, or at least had lost control of the nerve agent novichok which scientists say is made only in Russia".
He said Charlie Rowley was "absolutely not the brother I know" and said he was still concerned after speaking to his brother on the phone.
Sturgess' inquest will be opened today in Salisbury and the hearing is expected to be adjourned to allow police inquiries to continue.
It is understood Ms Sturgess was exposed to at least ten times the amount of Novichok the Skripals came into contact with.
British investigators now believe that the Skripals were most probably poisoned by current or former agents of Russia's military intelligence service, also known as the GRU, The New York Times reported last week, citing unidentified intelligence officers from the United Kingdom and US.
Traces of the nerve agent were found in a small bottle in the Amesbury home of Mr Rowley, 45, who remains in a serious but stable condition at Salisbury District Hospital.
Officers warned that searches of properties in Salisbury and Amesbury could last months after recovering 400 exhibits, samples and items.
Sergei Skripal served in the GRU, which is also cited in charges by US officials related to alleged Russian hacking in the 2016 presidential election.