An activist shareholder group in Cuisine de France owner Aryzta has increased its stake in the company to over 20% from just under 18% previously.
Swiss group Veraison and Spain's Cobas want the board of the Dublin listed food group to call an extraordinary general meeting of shareholders.
They wish to remove a number of directors, including chairman Garry McGann.
Cobas is proposing to replace the board members with its own candidates and has called for chief executive Kevin Toland to step down from the board to enable him to focus on its CEO functions.
Cobas also wants board members Annette Flynn, Dan Flinter and Rolf Watter to step down.
Aryzta announced a strategic review of operations in May and said it had appointed consultants to review the strategic and financial options available to it to maximise its value for the benefit of shareholders.
Rothschild & Co is carrying out the review, which is expected to be concluded by the end of July.
The move follows concerns among shareholders about ongoing falls in the company's share price.
Earlier this week, the Swiss and Dublin listed food group said it was seeing steady improvement in the revenue trend of the business.
But it cautioned that it expects a "bumpy" recovery over the coming months due to the coronavirus outbreak.
In a trading update, the baked goods group said that revenue for the month of June was down 23%. This compared with its revenues slumping by 36% in May and sinking 49% in April.
From the middle of March onwards, Aryzta said it took decisive and rapid action to protect and maximise liquidity in the business.
This included pausing production in bakeries to reduce capacity in line with demand, temporary laying off some staff, availing of government relief initiatives, suspending capital expenditure and eliminating discretionary cost to the maximum extent possible.
"These actions are being kept under constant review as the group expects a bumpy recovery over the coming months," Aryzta said.
It added that it continues to be vigilant with the impact and developments of Covid-19.
Shares in the company moved higher in Dublin trade today.