BAGHDAD (Reuters) - Kurdish authorities in Iraq offered on Wednesday to put an independence drive on hold, stepping up efforts to resolve a crisis in relations with Baghdad via dialogue.

But an Iraqi military spokesman suggested an offensive - launched to wrest back territory after Kurds voted overwhelmingly for independence in a referendum in September - would continue regardless.

The Iraqi government has transformed the balance of power in the north of the country since launching its campaign last week against the Kurds, who govern an autonomous region of three northern provinces and had held a swathe of other territory.

“The fighting between the two sides will not produce a victory for any, it will take the country to total destruction,” the Kurdistan Regional Government (KRG) said in a statement.

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

Baghdad has always considered the Kurdish secession referendum illegal. It responded last week by seizing back the city of Kirkuk, the oil-producing areas around it and other territory that the Kurds had captured from militant group Islamic State.

In a brief social media comment hinting that the campaign would continue, an Iraqi military spokesman said: “Military operations are not connected to politics.”

Prime Minister Haider al-Abadi has said the KRG should cancel the vote’s outcome as a pre-condition for talks. Several Shi’ite members of parliament on Wednesday asked him to stick to his position and not to accept just a freeze of the referendum.

On Wednesday, Abadi began an official visit to neighboring Turkey and Iran. Turkey and Iran both have Kurdish populations of their own and have supported Baghdad taking a hard line on the referendum.

Now that the Kurds have swiftly yielded most of their territory outside of their autonomous region to the advancing Iraqi forces, there are signs of pressure easing. Iran announced the reopening of one of the border crossings with the Kurdistan region, which it had closed earlier this month in support of the Iraqi government.

In Ankara, President Tayyip Erdogan said Turkey was ready to give all support to Baghdad as it seeks to reopen a crude oil pipeline from the Kirkuk oilfields to Turkey. The pipeline goes through the Kurdish autonomous region. Iraq had stopped shipping oil through it when the Kurds took Kirkuk in 2014.

In a statement to media after meeting Abadi, Erdogan said they discussed what political, military and economic steps they could take after what he called the “illegitimate” Iraqi Kurdish referendum last month.

STRATEGICALLY VITAL

Abadi has ordered his army to recapture all disputed territory and has demanded central control of Iraq’s border crossings with Turkey, which are all located inside the Kurdish autonomous region.

Kurdish Peshmerga forces beat back an advance by Iranian-backed pro-government paramilitaries on Tuesday in the region of Rabi’a, 40 km (25 miles) south of the Fish-Khabur border area with Turkey and Syria, Kurdish officials said.

Fish-Khabur is strategically vital because oil from both Kurdish and government-held parts of northern Iraq crosses via a pipeline there into Turkey, the main route out of the area for the exports that are crucial to any Kurdish independence bid.

The fighting so far has taken place outside the Kurdish autonomous region, but Fish-Khabur is inside it.

A boy rides a bicycle with the flag of Kurdistan in Tuz Khurmato, Iraq September 24, 2017. REUTERS/Thaier Al-Sudani

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.

U.S. Ambassador Douglas Silliman “worked with Iraqi security forces and Kurdish Peshmerga to end clashes endangering a Christian community and other civilians” north of Mosul, Washington’s Baghdad embassy said on its Twitter account.

The Iraqi government’s advance over the past week has been achieved with comparatively little violence, with Kurds mostly withdrawing without a fight.

But it caused Kurds to flee en masse from Tuz Khurmato, a multi-ethnic city south of Kirkuk where sectarian tension flared after Iraqi forces took control, humanitarian organizations said.Most of those displaced, estimated at nearly 30,000, are in need of urgent aid and staying in open shelters, officials from two international humanitarian organizations told Reuters.

Abadi pledged in an interview to the Wall Street Journal to disarm Shiite militias that refuse to come under his control after Islamic State’s defeat.

Iraqi forces are preparing in parallel an offensive to recapture the last patch of Iraqi territory still in the hands of Islamic State, on the border with Syria, the military said on Wednesday.

The militant group also holds parts of the Syrian side of the border, but the area under their control is also shrinking there as they retreat in the face of two sets of hostile forces - a U.S.-backed, Kurdish-led coalition and Syrian government troops with foreign Shi’ite militias backed by Iran and Russia.