Paul Kelly, deputy leader of North Lanarkshire Council, was speaking the day after plans for 500 redundancies at the Uddingston base were made public.
Belgium owners Ageas said it was looking to close the site by the end of March next year.
The Scottish government said it was "ready to assist" if jobs do go.
Mr Kelly told BBC Radio Scotland's Good Morning Scotland programme that closure would have a "direct impact on the Lanarkshire area and the local community".
He added: "This is a very significant number of jobs and our immediate concern is with the workforce and their families. This is the lead up to Christmas, it is devastating news.
"But we are wanting action, we are wanting to immediately act on this that is why we have called on the establishment of a taskforce."
Mr Kelly urged the Scottish government to get round the table with other partners "to do all we can" and "as soon as possible".
Scottish Enterprise and the Scottish government are understood to be seeking another company which could use the insurance firm's contact centre skills.
Five years ago, the company had more than 900 staff in North Lanarkshire. It now employs 521 people at the Tannochside Business Park site.
An Ageas spokeswoman said Kwik Fit Insurance Services had been "under pressure for some time" as a result of changes in the way people buy insurance and the way the personal insurance market now operates.
Mr Kelly said he did "not disagree" with the reasons Ageas had given for the planned closure but he believed the current workforce was "highly skilled and a huge asset" to the wider business community.
He explained that the council had written to First Minister Nicola Sturgeon asking for the establishment of a taskforce.
Business minister Paul Wheelhouse said on Tuesday that he was "very disappointed" to learn of the proposed closure.
He added: "Should any job losses proceed, we stand ready to assist those affected through our initiative for responding to redundancy situations, Partnership Action for Continuing Employment (PACE) and our Finance Sector Jobs Taskforce.
"I am keen to meet with North Lanarkshire Council to work together with them to ensure we support the local community and secure investment at the site."
Ageas said it would now enter into a consultation period ahead of the planned shut down next year.

The target was reached eight weeks after Mr Aikman's death on 2 February at the age of 31.
Now the Gordon's Fightback campaign is seeking to raise £1m in his memory to fund research into the disease.
Mr Aikman was director of research for the Better Together campaign at the time of his diagnosis.
His campaign has been supported by leading figures in all of Scotland's major political parties.
Gordon's husband Joe Pike said: "I miss Gordon's voice, his eyes and his smile. I miss the in-jokes and affection and finishing each other's sentences.
"But Gordon never wanted pity, and I don't either. I want to turn a negative into a positive. Today, I want everyone to mark Gordon's birthday by giving the gift of a cure for Motor Neurone Disease.
"Gordon's real passion was finding a cure. Doctors do not know what causes Motor Neurone Disease, let alone how to cure it. But scientists in labs doing research provided him with hope. Not that he would be saved, but hope for future generations.
"A cure for MND - that would be a real gift to remember Gordon by."
A taskforce to save jobs at the Kwik Fit insurance office in Lanarkshire should be set up "as soon as possible", a local councillor has said.
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The fund named after Scottish motor neurone disease (MND) campaigner Gordon Aikman has raised £600,000.