BAGHDAD (Reuters) - Iraq’s prime minister demanded on Thursday that Kurds declare their independence referendum void, rejecting the Kurdish autonomous region’s offer to suspend its independence push to resolve a crisis through talks.

Iraq's Prime Minister Haider al-Abadi speaks with U.S. Secretary of State Rex Tillerson (not pictured) in Baghdad, Iraq October 23, 2017. REUTERS/Alex Brandon/Pooll

“We won’t accept anything but its cancellation and the respect of the Constitution,” Haider al-Abadi said in a statement during a visit to Tehran.

The Kurdistan Regional Government (KRG) proposed on Wednesday an immediate ceasefire, a suspension of the referendum result and “starting an open dialogue with the federal government based on the Iraqi Constitution”.

The Kurds have been swiftly making concessions to Baghdad since last week, when Abadi sent his forces to seize all Kurdish-held territory outside of three autonomous provinces.

A startlingly rapid advance by government troops transformed the balance of power in northern Iraq within a matter of days and has wrecked decades-old dreams of Kurdish independence that had come to a head last month with a referendum on secession.

Baghdad has always considered the Kurdish secession referendum illegal. Abadi’s visit to Iran on Thursday follows a trip to Turkey on Wednesday, a diplomatic offensive that has shored up support from Iraq’s neighbors for his hard line.

Turkey’s foreign minister Mevlut Cavusoglu said on Thursday the Kurdish offer to suspend the push for independence was a step in the right direction but did not go far enough.

Abadi has ordered his army to recapture all disputed territory and has demanded central control of Iraq’s border crossings with Turkey, including the oil export pipeline hub at Fish-Khabur, located just inside the Kurdish autonomous region.

A media assistant to Kurdish leader Masoud Barzani said Kurdish security forces, known as Peshmerga, had repelled three attacks by Iraqi forces, two in the direction of Fish-Khabur, and one in Perde, on the road linking Kirkuk to the KRG capital Erbil, destroying several tanks and armored vehicles.

Iraqi authorities did not confirm this account of fighting. Both contested areas have important oil assets.

IRAN SEEKS TO DRIVE WEDGE

The fighting between the central government and the Kurds is particularly tricky for the United States which is a close ally of both sides, arming and training both the Kurds and the central government’s army to fight Islamic State.

Iraq is one of the only countries in the world that is closely allied to both the United States and Iran, and Tehran has used the Kurdish separatism bid as a way to drive a wedge between Washington and Baghdad.

Iran’s supreme leader, Ayatollah Ali Khamenei told Abadi in Tehran that he should not rely on the United States in the fight against Islamic State.

“Unity was the most important factor in your gains against terrorists and their supporters,” Khamenei said, according to state TV. “Don’t trust America ... It will harm you in the future.”

Last week, Abadi spurned a call from U.S. Secretary of State Rex Tillerson to send “home” Iranian-backed paramilitary fighters. Abadi’s office called paramilitaries “patriots”.

The Kurdish crisis has broken out even as Iraq is about to finally defeat Islamic State, also known as ISIL, ISIS or Daesh, after a three-year war in which it received strong backing from a U.S.-led coalition, the Kurds and Iran.

Iraqi forces launched an offensive on Thursday to recapture the last patch of Iraqi territory still in the hands of Islamic State, on the border area with Syria.

“Daesh members have to choose between death and surrender,” Abadi said in a statement announcing the offensive on region of al-Qaim and Rawa.

Islamic State’s self-declared cross-border caliphate effectively collapsed in July, when U.S.-backed Iraqi forces captured Mosul, the group’s de facto capital in Iraq, in a grueling battle which lasted nine months.

The militant group still holds parts of the Syrian side of the border, but the area under its control has rapidly shrunk there too, with its de facto Syrian capital falling to a U.S.-backed, Kurdish-led force last week.

Regular Iraqi army units, Sunni tribal forces and Iranian-backed Popular Mobilisation are taking part in the offensive toward the Syrian border, Iraq’s Joint Operations Command said.