Dollar General's offer of $78.50 per share in cash represents a premium of 3.2 percent to Family Dollar's Friday close. Family Dollar had 113.9 million shares outstanding as of July 5, according to Thomson Reuters data.
Family Dollar's shares rose 5.1 percent to $80 before the bell, while Dollar General's shares rose 11 percent to $63.90.
The deal, at an enterprise value of $9.7 billion, was proposed in a letter to Family Dollar's board on Monday.
Dollar Tree offered last month to pay $8.5 billion, or $74.50 per share in cash and stock, for Family Dollar, representing an enterprise value of about $9.2 billion.
Billionaire activist investor Carl Icahn had pushed Family Dollar to sell itself to Dollar General.
Dollar General said a deal would create a "preeminent small-box retailer," with nearly 20,000 stores in 46 U.S. states and sales exceeding $28 billion.
Chief Executive Rick Dreiling said the "financially superior" offer would provide Family Dollar's shareholders with a substantial premium and immediate liquidity.
Dollar stores have struggled in a weak U.S. economy as Wal-Mart Stores Inc <WMT.N> and other large retail chains chase penny-pinching consumers by offering more items priced at $1 or less.
Most Family Dollar stores are in low-income neighborhoods. Its presence is biggest in Texas and the eastern United States, where it mainly sells lower-margin food and household products.
Dollar General also said it was prepared to divest up to 700 retail stores to get regulatory approval for the deal. This is the same percentage that Dollar Tree proposed in its deal to buy Family Dollar.
Dollar General said it had financing from Goldman Sachs and Citigroup Global Markets Inc for the deal, including $305 million termination fee payable to Dollar Tree.