A consortium of Doyle family members and two Jurys Doyle directors have approached the company about a possible takeover.
The group of five - Eileen Monahan, Ann Roche, Bernadette Gallagher, Walter Beatty and Elizabeth Nelson -have a shareholding of 42.3% in the company.
Any deal would guarantee a minimum share price of no less than €18.90 (£12.72) per share, valuing the company at €1.19b (£801m).
It leaves property developer Sean Dunne as the other major shareholder, with a 29.9% shareholding in the company.
One insider close to the deal said it was likely the Doyles would come to an agreement with Dunne before the extraordinary general meeting scheduled for 27 September.