Dollar General Corp raised its bid for Family Dollar Stores Inc to $80 per share, or $9.1 billion, and warned it may turn hostile and appeal directly to shareholders if the new offer was rejected.

The No.1 U.S. deep-discount retailer also said it would pay a break-up fee of $500 million if the deal ran foul of competition law, the reason Family Dollar had cited for its rejection of the earlier $8.95 billion offer.

Family Dollar opted instead for an $8.5 billion cash-and-stock bid from Dollar Tree Inc.

Family Dollar's shares were up almost 1 percent at $80.52 in premarket trading on Tuesday, above Dollar General's offer, suggesting some investors held hopes of higher offers.

Shares of Dollar General, which increased its all-cash offer by 2 percent, were up 2.3 percent.

"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 proposed earlier.

The company said it hired Richard Feinstein, a former director of the Bureau of Competition at the Federal Trade Commission, to further "validate its antitrust analysis".

Playing down antitrust concerns, Dollar General said its documents would show that the products it offers are not unique as they are also available at most mass retailers, drug and grocery stores.

It also said that Wal-Mart Stores Inc, and not Family Dollar, was the primary driver for the company's pricing decisions.

Dollar General and Family Dollar offer goods at multiple price points, while Dollar Tree sticks to a $1 or less format.

The Dollar Tree bid is friendlier to Family Dollar's management.

Chief Executive Howard Levine, son of Family Dollar's founder, would remain CEO of the Matthews, North Carolina-based company were it to be bought by Dollar Tree.

If Dollar General succeeds in buying Family Dollar, Levine is widely expected to lose his job, although this has not been confirmed.

Family Dollar was not immediately available for comment.