Company owner Ageas said it is “under pressure” due to “fundamental” changes in the way people now buy insurance.
Ageas has now started a 45-day consultation with the 521 staff over plans to close the office in Uddingston, South Lanarkshire, next year.
It is also working with Scottish Enterprise and Scottish Development International in an attempt to find “third parties who may be able to use the facility and skills available at the site”.
An Ageas spokeswoman said:”Kwik Fit Insurance Services has 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.
“We have made a number of attempts to address this but they have not been sufficient to improve performance and address declines in workload.
“This means difficult choices have to be considered. Subject to employee consultation, we are proposing to close our Uddingston office next year.
“This is clearly very difficult news for our employees and is not a course of action we have taken lightly. Our priority is to support them through what we know is a very unsettling time.
“We are committed to listening to them, and being open and transparent during the consultation.”
Ageas said the Brexit vote or a potential second Scottish independence referendum were not factors in the decision, and it is simply down to “fundamental changes” in the personal insurance market.
Ageas bought Kwik Fit Insurance Services in 2010 from Kwik Fit GB.
Local SNP MSP Richard Lyle said: “Understandably this will be a very difficult time for the workforce and everyone affected.
“I have been in contact with Ageas to ask that they ensure staff are meaningfully included in the consultation and that uncertainty over their future is kept to a minimum, and will be visiting the workforce on Friday to provide support.
“The Scottish Government is actively involved in providing all necessary support for concerned individuals through the Pace initiative, and I would encourage members of the workforce with concerns or who may be looking for advice to use this service or to contact me as their local MSP.”