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The fight for one of America’s biggest deep-discount stores isn’t over who is willing to pay top dollar, at least for now.

Family Dollar’s decision on Thursday to reject an $8.9 billion takeover bid by its bigger competitor, Dollar General, in favor of an earlier $8.5 billion deal with a smaller rival, suggests that a bruising battle looms.

Though it may seem strange for a would-be seller to spurn a higher-priced offer, Family Dollar argued that its earlier agreement, with Dollar Tree, was more likely to actually get done. A sale to Dollar General would instead present what Family Dollar argued would be significant antitrust problems that could derail any deal.

In its $78.50 a share offer, Dollar General said that it would be willing to sell as many as 700 stores to win approval for any deal. But Family Dollar is arguing that such a maneuver would not be enough to satisfy government regulators and that it would be better off accepting the $74.50 a share that Dollar Tree will pay.

Behind the differences in opinion are contrasting views about how the Obama administration would view the industry where the three companies compete. To Family Dollar, that is the dollar-store sector specifically, where it ranks second and Dollar General first. To Dollar General, the range of competitors is far broader, encompassing stores like Walmart and Walgreens.

By Thursday afternoon, Dollar General and Family Dollar appeared set for a potential fight. Dollar General said in a statement that it was “disappointed” by the rejection and was reviewing its options, reiterating that it believed its bid was superior to Dollar Tree’s.

“It is clear through the most recent releases that we’ve seen that Dollar General is very serious about getting this deal done,” Paul Trussell, an analyst at Deutsche Bank, said by phone. “I would not be surprised to see Dollar General present a tender offer to Family Dollar shareholders themselves.”

Shares in Family Dollar closed at $79.41, above both takeover bids.

Family Dollar’s swift rejection came less than 12 hours after Dollar General accused its target’s chief executive, Howard R. Levine, of breaching a fundamental duty of most corporate executives: maximizing value for shareholders. Underpinning Mr. Levine’s lack of interest in a deal, according to Dollar General, is his fear that he might lose his job.

In Family Dollar’s long rejection notice, Mr. Levine and several board members argued that their basic concern was much simpler: Combining the country’s two biggest dollar stores would pose serious risk of rejection from antitrust regulators.

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Family Dollar acknowledged that it had heard several times from Dollar General about a possible merger over the last year and a half. But it said Dollar General had canceled proposed meetings several times, including one scheduled for June to discuss antitrust issues.

According to Family Dollar, by the last time the two sides met, on June 19, Dollar General expressed no interest in pursuing a deal. And Family Dollar had by then signed a nondisclosure agreement with Dollar Tree that prevented any mention of their merger talks.

The proposed merger of Family Dollar and Dollar Tree was announced just more than a month later.

Family Dollar said its decision to reject the latest offer was backed by four independent directors, including Edward P. Garden, an executive at Trian Fund Management, the hedge fund run by Nelson Peltz, which owns a 7.3 percent stake.

Support by Mr. Peltz’s firm for the Dollar Tree bid may blunt criticism from Family Dollar shareholders that the company is ignoring a more lucrative bid.

“Given the significant antitrust issues involved with Dollar General’s proposal, we will not jeopardize the Dollar Tree deal for a transaction with Dollar General that has a high likelihood of not closing due to antitrust considerations,” Mr. Garden said.

The billionaire Carl C. Icahn, who also owns a big stake in Family Dollar, has accused Mr. Levine and his management team of not looking out for the best interests of shareholders in their decision.

The gap between the two companies may not be unbridgeable. Family Dollar is still demanding stronger protections against antitrust risk from its bigger competitor, according to a person briefed on the matter. Among those may be a so-called hell or high water provision that would compel Dollar General to do whatever is necessary, including divesting more stores, to satisfy regulators.

Dollar General was noncommittal about what it might eventually agree to do. “We remain willing to share this analysis with Family Dollar and its counsel and are confident that we will be able to quickly and efficiently resolve any potential antitrust issues,” the company said in a statement.

Morgan Stanley and the law firm Cleary Gottlieb Steen & Hamilton are advising Family Dollar.