A person familiar with the deal told The Associated Press on Tuesday night that the contract runs through the 2014 season. The person spoke on condition of anonymity because the deal was pending a physical and hadn't been finalized.
The agreement came two days before a scheduled arbitration hearing.
The 23-year-old Andrus is a .271 career hitter with 102 stolen bases and has played at least 145 games in each of his three seasons for the Rangers. The new deal will take Andrus through his final three arbitration-eligible seasons.
Andrus hit a career-best .279 with 37 stolen bases last year when the Rangers won their second consecutive American League pennant. He had asked for $3.6 million in arbitration and was offered $2.65 million for next season before agreeing to a multiyear deal with Texas.
Cruz, whose eight postseason homers last year included six in the AL championship series with a game-ending grand slam in Game 2, asked for $7.5 million while the Rangers offered $5.5 million. He made $3.65 million last year, when he hit .263 with 29 homers and 87 RBIs in 124 games.
Napoli set career highs by hitting .320 with 30 homers and 75 RBIs in 113 games in his first season with Texas, when he made $5.8 million after spending his first five seasons with the Los Angeles Angels. He asked for $11.5 million, and Texas offered $8.3 million.
Andrus was among several key players the Rangers got from Atlanta in a July 2007 trade for Mark Teixeira after the switch-hitting first baseman turned down a contract extension.
Andrus made his major league debut in the 2009 season opener after the Rangers moved five-time All-Star shortstop Michael Young to third base for the shortstop who was then only 20 years old and had never played above the Double-A level.