Dollar General also said it would pay a break-up fee of $500 million if deal ran foul of competition law, the reason Family Dollar had cited for its rejection the earlier offer and opting instead for an $8.5 billion cash-and-stock bid from Dollar Tree Inc (DLTR.O).
Family Dollar's shares were up almost 1 percent at $80.50 in premarket trading, above Dollar General's cash offer, suggesting some investors held hopes of higher offers.
"In the event you refuse to engage with us regarding our revised proposal, we will consider taking our persuasive and superior proposal directly to your shareholders," Dollar General Chief Executive Rick Dreiling said in a letter to Family Dollar's board on Tuesday.
Dollar General also said it was willing to sell up to 1,500 stores to clear the antitrust review, up from the 700 it had said earlier.
Dollar General said last week it remained committed to acquiring Family Dollar, after the target rejected the previous offer, saying the deal could run foul of competition law. Instead, it opted for a lower bid from Dollar Tree Inc (DLTR.O).
Family Dollar was not immediately available for comment.
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