Iraq's pro-Iran armed faction Kataeb Hezbollah, accused by Israel of holding an Israeli-Russian academic, implied on Thursday that it was not involved in her disappearance.
The group said it was doing everything it could to uncover the fate of "Zionist hostage or hostages" in the country.
Kataeb Hezbollah is a powerful faction of Iraq's Hashed al-Shaabi, Iran-backed former paramilitaries that were integrated into the Iraqi security forces in recent years.
In the group's first reaction to the Israeli accusation on Thursday, its spokesman Abu Ali al-Askari issued an ambiguous press release on Telegram.
"The admission by the prime minister of the Zionist entity (Israel) concerning the presence of an Israeli security agent hostage in Iraq is a very dangerous indicator," it said.
"The security organisations concerned should expose networks linked to this entity and bring them to justice," it added.
"In turn, Kataeb Hezbollah will work tirelessly to find out the fate of the Zionist hostage or hostages in Iraq... to discover more about the intentions of the criminal gang which facilitates their movements in a country... that prohibits and criminalises any relationship" with Israel.
The statement from Netanyahu's office said Tsurkov had visited Iraq "on her Russian passport at her own initiative pursuant to work on her doctorate and academic research on behalf of Princeton University in the US".
According to an Iraqi intelligence source, Tsurkov was kidnapped in Baghdad "at the beginning of Ramadan", the Muslim fasting month which this year started on March 23.
She had arrived in Baghdad "at the beginning of December 2022", a Western diplomat stationed in Iraq said on condition of anonymity.
Tsurkov's website said she was a fellow at the New Lines Institute for Strategy and Policy, as well as a research fellow at the Forum for Regional Thinking, an Israeli-Palestinian think-tank based in Jerusalem.
"Liz is very much a part of America. She works with a Washington think tank, writes for an American magazine and studies at Princeton University. She deserves America's every effort to bring her to safety," New Lines said on Wednesday.
Witnesses who knew her said Tsurkov used to freely move around in the Iraqi capital.
She had focused on pro-Iran factions and the movement of Iraqi Shiite leader Moqtada Sadr as part of her research on the region, according to several journalists who had met her.