The revelations may have been brushed off by Benjamin Netanyahu, but they have the potential of developing into a serious new political scandal.
Israeli defence sources had earlier confirmed to local journalists that accusations made by the opposition politician Avigdor Lieberman were correct.
Lieberman, who heads the Yisrael Beiteinu party, had told the public broadcaster, Kan, that Netanyahu had unilaterally approved the transfer of weapons to the Abu Shabab clan.
"The Israeli government is giving weapons to a group of criminals and felons, identified with the Islamic State group," Lieberman said.
"To my knowledge, this did not go through approval by the cabinet," he added.
Defence sources subsequently confirmed that Israel had been arming the Abu Shabab clan with Kalashnikov rifles, including some that had been seized from Hamas.
The clan has been operating in Rafah, in an area under Israeli military control.
However, Yasser Abu Shabab posted online to "categorically reject" that Israel had supplied his group's weapons.
"Our weapons are simple, outdated and came through the support of our own people," he said.
Hamas sources have said that Abu Shabab's activities have become an issue - with one report in an Arabic newspaper saying that the Hamas armed wing had begun carrying out assassinations of members of the clan.
Netanyahu's office said that Israel "was working to defeat Hamas through various means, based on the recommendations of all the heads of the security establishment".
Strong criticism of the initiative has come from Yair Golan, leader of the Democrats in the Knesset.
In a social media post on X he said: "Netanyahu is a threat to Israel's national security. Instead of bringing about a deal...bringing the hostages home and providing security for Israeli citizens, he is creating a new ticking time-bomb in Gaza".
