Yoplait is the world's second-largest yogurt maker and is owned by French investment firm PAI partners and French cooperative dairy group Sodiaal.
If successful, General Mills would acquire the roughly 50 percent stake held by PAI partners and work with Sodiaal, which is retaining its stake.
General Mills did not disclose the value of the offer but the Wall Street Journal, citing a personal familiar with the situation, has placed it at $2.2 billion.
The acquisition would build on General Mills' aim to increase its health and wellness product portfolio and expand operations in France, where it already produces several products.
General Mills, based in Minneapolis, already has a partnership with the company. It has licensed the Yoplait brand for more than 30 years, helping it build into a top yogurt brand in the U.S., one Yoplait's largest markets.
The company said discussions are in progress and that it is also initiating talks with French works councils.
Yoplait officials in France said they won't comment on the negotiations until they are finished talking to the works council and other concerned parties, which they predicted would be the middle of next week.
At $2.2 billion, the offer would beat out reported competitor's bids including Bright Foods of China, Nestle SA of Switzerland, Mexico's Grupo Lala Axa Private Equity and French dairy company Groupe Lactalis.
Shares of General Mills, one of the world's largest food companies, rose 59 cents to $36.72 in midmorning trading.
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