The deal, which will cover all three of Andrus' arbitration years, is expected to be in the $14 million to $15 million range.
The contract does not include any club options and is contingent on Andrus passing a physical, sources said.
Andrus, represented by Scott Boras, will be eligible for free agency at age 26 after his new deal expires.
The agreement enables the Rangers to avoid arbitration with Andrus, a defensive stalwart who helped the team win back-to-back American League titles.
Andrus, 23, had asked for $3.6 million in arbitration. The Rangers had offered $2.65 million. A hearing was scheduled for Thursday.
Andrus reached career highs in virtually every offensive category last season, batting .279/.347/.361 with five home runs, 60 RBI and 37 stolen bases in 49 attempts.