Laing O'Rourke is understood to have held talks over a potential takeover of parts of its business.
Delegates from the construction giant spent two days last month at the Quinn headquarters, in Co Cavan, it is reported.
Soon after the bid became public, it emerged that Group's founder, Sean Quinn was to stand down from the board and that a Stormont nominee was to step in to co-ordinate the NI Government's response to the situation.
The Fermanagh businessman said this week that his decision was based on his need to "concentrate in the short term on family interests outside of Quinn Group" as both he and his wife Patricia stepped back from their board positions.
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The NI Enterprise Minister Arlene Foster and Employment and Learning Minister, Sir Reg Empey have now jointly announced the appointment of Leslie Ross, to look after Quinn from their perspective.
Last week, plans to sell the insurance arm were announced and the news came that administrators were forcing job losses, hitting more than a third of the company's 2,400-strong workforce.
Meanwhile, Laing O'Rourke is well placed for a buy-out as the largest privately owned construction solutions provider in the UK.
Formed over 30 years ago, the Group's international operations posted managed revenue of £5bn in 2009.
The business is made up of a range of engineering, construction and specialist services companies, which combine to provide prestigious clients with a comprehensive investment, development and management capability.
It employs over 30,000 people across Europe, the Middle East, South Asia and Australasia, with operations that span five core sectors: lifestyle; business; social infrastructure; transport and mining; and energy, utilities and waste.
The firm is looking at 'cherry-picking' the best bits of the Quinn Group.
Quinn Insurance, formerly the most lucrative part of the business portfolio, was put into administration after the financial regulator said that it had breached key rules governing the insurance market.
See: Quinn Steps Back As Stormont Steps In
(BMcC/GK)
