The Israeli hackers are described as having spotted Russian hackers searching for NSA hacking tools after they themselves breached Kaspersky's internal systems.
Israeli spies warned USA intelligence that antivirus software employed by several American government agencies was being used by Kremlin-backed hackers to steal secrets (write Boer Deng and Anshel Pfeffer).
On Wednesday, The New York Times cited multiple people who have been briefed on the matter in a report claiming that it was Israeli hackers who discovered Russian hackers using the anti-virus software to steal American hacking tools.
The U.S. government last month ordered Kaspersky software removed from government computers, saying it was concerned the Moscow-based cyber security firm was vulnerable to Kremlin influence.
The Russian operatives were reportedly using a Kaspersky antivirus program, which is widely used to scan a computer for malicious files, as a quasi-search engine to scan for codenames of USA intelligence programs.
Several cyber security specialists believe the FSB, the Russian intelligence service, has exploited a loophole of Kaspersky software to gain access to their customer's computers.
That access, it concluded, could help enable cyber attacks against U.S. government, commercial and industrial control networks, the Post reported.
The Times reported that Russian hackers turned the Kaspersky software into a sort of Google search for sensitive information, and that it is not yet publicly known what other USA secrets they may have discovered.
The New York Times said that the Russian embassy had not responded to a request for comment.
The NSA, the White House and the Israeli embassy in Washington have not commented on the matter.
Kaspersky has said it was neither involved in nor aware of the situation and denies collusion with authorities.
Eugene Kaspersky, a businessman and former Russian Defense Ministry official, founded Kaspersky Lab in 1997.
United States intelligence agencies have concluded that Russian President Vladimir Putin ordered a multipronged digital influence operation past year in an attempt to help Donald Trump win the White House, a charge Moscow denies.