Shares of Cossette Inc were up 1.8 percent at C$7.94 in morning trading after rising as much as 4.5 percent earlier.
Cosmos, led by two former Cossette founders, offered late on Tuesday to pay C$131.5 million ($125.1 million), or C$7.87 a share, for the company and said it could bid even higher.
The offer matched a November 10 bid from U.S. private equity firm Mill Road Capital, securing lock-up agreements Cosmos has with holders of about 18.7 percent of Cossette stock. Cosmos itself already holds another 18.7 percent of Cossette.
Cossette, relatively small by international standards, is a giant in Canada, where it grew from a tiny shop in the early 1970s to a transnational company with clients including McDonald's Restaurants of Canada, Bell Canada, General Motors of Canada and Coca-Cola Ltd.
(Reporting by Pav Jordan; Editing by Lisa Von Ahn)