NEW YORK (TheStreet) -- Dollar General (DG) didn't announce any new offer for Family Dollar (FDO)  as the discount retailer reported mixed second-quarter earnings on Thursday.

Dollar General CEO Rick Dreiling did say he remains intent on a merger and the company is reviewing its options. One source familiar with Dollar General's thinking conceded on Thursday that the ball remains in the company's court after its initial efforts at a merger were rejected by Family Dollar in mid-August. Dollar General has offered $78.50 a share in cash for Family Dollar, topping a $74.50 cash-and-stock merger that was agreed between the company and Dollar Tree (DLTR) in late July.

Family Dollar didn't engage Dollar General's higher offer in earlier this month, stating that antitrust hurdles presented too much of a risk to its shareholders. With hedge fund investors such as Trian Management privy to and supportive of Family Dollar's sale process, it will be interesting to see what strategy Dollar General uses in trying to push its bid forward.

Read More: Dollar General Affirms Interest in Family Dollar Merger

There are some obvious ways for Dollar General to keep the ball rolling on its competing offer for Family Dollar. While Dollar General is already offering a higher price than Dollar Tree, it could increase that offer to create a new formula of risk-reward for Family Dollar's board to consider. BMO Capital Markets analysts also believe Dollar General could offer some assurances to mitigate antitrust risks.

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