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Dollar General is willing to pay more to win the bidding war over Family Dollar Stores. And its takeover effort could go hostile if Family Dollar does not begin sale negotiations.

Dollar General offered on Tuesday to raise its bid for Family Dollar to $9.1 billion after the rejection of an earlier offer that would have beaten a rival $8.5 billion deal with Dollar Tree, agreed to in late July.

Perhaps more important, Dollar General — the biggest dollar-discount chain in the country — also added new provisions to its bid to provide more assurances to Family Dollar Stores in case their proposed merger runs into antitrust issues.

In rejecting Dollar General’s original bid last month, Family Dollar publicly cited antitrust concerns as a chief reason, and privately suggested that any such bid needed stronger assurances — such as what is known as a “hell or high water” provision requiring any and all reasonable steps to guarantee a deal’s closing — should the government closely scrutinize the proposed union.

Yet if Family Dollar does not begin talks soon, Dollar General warned that it might take its case directly to its target’s shareholders, raising the possibility of a bitter fight between two of the country’s biggest deep-discount retailers.

The move increased the stakes for Family Dollar and its investors, including two prominent hedge fund managers — the billionaires Carl C. Icahn and Trian co-founder Nelson Peltz — who have argued that the company has not done enough to improve returns for shareholders.

It may also have put additional pressure on Dollar Tree, which is smaller than the other companies, to offer even more if it hopes to remain a competitive bidder.

Behind the dueling merger offers is an effort to respond to an economic recovery that has eluded many middle-class Americans, many of whom have since looked to such stores for bargain-hunting. A union of Family Dollar with Dollar General would create a 20,000-store giant with more than $28 billion in sales, better able to compete with the likes of Walmart.

A combination of Family Dollar and the smaller Dollar Tree, on the other hand, would create a more significant rival to Dollar General.

Even Edward P. Garden, Trian’s chief investment officer and a board member, said in a statement at the time that he would not throw away a sure deal for a potential transaction with a much higher likelihood of failure.

Dollar General has argued that any deal between it and Family Dollar would clear regulatory review, since the two companies compete with a host of retailers like pharmacy chains, convenience stores, Walmart and Amazon.

The new offer unveiled on Tuesday was meant to defuse every objection that the target of its takeover interests had raised about a potential tie-up of the two.

“We’re trying really hard to get some engagement here,” Richard W. Dreiling, Dollar General’s chief executive, said in an interview on Tuesday. Speaking about the new bid, he added, “It’s a fantastic offer designed to bring their board to the table to talk with us.”

Under the terms of the revised offer, Dollar General says it would pay $80 a share, up from its original bid of $78.50 a share. Both offers exceed Dollar Tree’s $74.50-a-share bid. The latest bid is nearly 32 percent higher than where Family Dollar’s shares were trading before the Dollar Tree deal was announced.

The crux of the new bid lies in measures aimed at protecting Family Dollar in case the Obama administration takes a harsh look at any deal. Dollar General agreed to pay a $500 million reverse termination fee if a merger falls apart over antitrust concerns.

To further address competition issues, Dollar General has agreed to more than double the number of stores it is willing to sell off to win regulatory approval, to 1,500. Even so, Mr. Dreiling argued, any combination would pass muster with far fewer sell-offs.

Family Dollar said in a statement that it was reviewing its competitor’s latest proposal, though its board did not change its recommendation of the Dollar Tree agreement. A spokesman for Dollar Tree declined to comment, while Mr. Garden was not available for comment.

Shares in Family Dollar rose slightly on Tuesday, to $80.22, suggesting investor belief that an even higher bid might eventually emerge. Shares in Dollar General also rose slightly, to $64.36, while those in Dollar Tree rose 1.6 percent, to $54.46.