NEW YORK – ConAgra Foods Inc. made a $4.9 billion bid for Ralcorp Holdings Inc., but the maker of Post cereal and store-brand products rejected it Wednesday.

ConAgra has made two offers to acquire RalCorp in an effort to expand its business into private-label products.

Ralcorp said the $86-per-share bid from ConAgra is not in shareholders’ best interests. It followed an $82-per-share offer in March that Ralcorp also refused.

The latest offer is a 3.2 percent premium over Ralcorp’s most recent closing stock price of $83.33 but well below what some analysts estimate it could fetch.

Ralcorp Chairman William Stiritz said the company has a proven track record of delivering shareholder value, and that the company is confident that its team and strategic plan will continue to do so as an independent business.

Ralcorp also said its board has adopted a shareholder rights plan, details of which are unavailable until it is filed with regulators. Such a plan would reduce the likelihood that a person or group could gather up shares to influence future decisions.

ConAgra’s brands include Chef Boyardee and Slim Jim. Closing the deal would make the company the third-largest packaged food maker in the U.S.

“Ralcorp has made significant progress with its businesses, and we are excited about the prospect of building on its number one position in private label and enhancing its iconic brands, like Post, in very important categories,” ConAgra CEO Gary Rodkin said in a written statement.

Jonathan Feeney of Janney Capital Markets said in a client note that moving to an all-cash offer was likely an attempt by ConAgra to make the proposal more attractive. But the $86-per-share offer is still below Feeney’s $98-per-share estimate for Ralcorp.

Shares of Ralcorp rose $4.06, nearly 5 percent, to close at $87.39 Wednesday. ConAgra’s shares rose 76 cents, or 3.1 percent, to $25.51.

ConAgra disclosed in a letter to Stiritz that it made a call to Ralcorp in February to try to arrange a meeting to talk about a potential takeover. The company went on to say that its initial offer made in March was rejected by Ralcorp in a phone call April 1 and a rejection letter Sunday.

The bid ConAgra submitted Wednesday also includes the assumption of $2.5 billion in debt. ConAgra, based in Omaha, Neb., says it would pay for the buyout with available cash and by issuing debt.

ConAgra said that combining its existing private-label business with Ralcorp’s operations would create about $4 billion in combined annual private-label sales.