The upper market retailer has two outlets in Scotland - one on Multrees Walk and the other in Princes Square Shopping Centre in Glasgow.
Both of those Scottish shopfronts could be set to shut.
The jeweller has appointed Deloitte as administrators in an attempt to secure a rescue sale, while the business continues to trade.
What does Links of London sell?
Links sells luxury British jewellery, watches, cufflinks and gifts from 28 stand-alone stores and seven concessions across the UK and Ireland.
Deloitte said the retailer has struggled to cope with "difficult trading conditions", which have weighed on a number of other UK high street chains.
The British brand is owned by troubled Greek company Folli Follie, which was plunged into crisis over a fraud related to overstating sales.
The owners considered a company voluntary arrangement (CVA) restructuring process, refinancing or a straight sale, but were unable to conclude a deal.
The administrators said that "in light of ongoing cash flow pressures", the firm's directors were left with "no choice" but to place it into administration.
Deloitte said the business intends to continue to trade while it seeks out sale options, but said it could sell stock and assets over a period of trading "for the benefit of the company's creditors".
Links, which was founded in 1990, saw sales slide 12 per cent to £42.9 million and plunged to a £20.5m pre-tax loss in its most recently filed accounts for the year to December 2017.
Mike Ashley linked to failing retailer
Vulture fund Hilco and Sports Direct tycoon Mike Ashley were linked to the retailer as it sought a buyer last month.
Matt Smith, joint administrator for Links of London, said: "The company is well-known in its market, having been present on British high streets for almost 30 years.
"This is not the outcome we hoped for and will of course be difficult news for employees and their families.
"We appreciate the support of management and we will continue to support employees through this time."
Why the high street is failing
Ian Pollard, vice-president of business transformation group Signavio, said: “There is no denying that the growth of online digital rivals has played a pivotal role in the decline of the British high street.
"E-commerce sales will likely account for 53 per cent of the entire retail market by 2028, which is due to faster and cheaper deliveries, and the array of personalised choice online.
"But retailers can’t just blame the rise in digital competitors for their sharp fall in profits. They are struggling to be agile in an increasingly digital market and fail to act on real-time intelligence, which can enable them to make smarter business decisions.
"Retailers need to address this crisis by taking a customer-first approach and having a holistic view of their processes that have the greatest effect on the customer.
"By ensuring that an effective customer journey sits at the forefront of all decisions made in these competitive environments will allow these companies to reverse the current decline.”