The sides hammered out the agreement during the past couple days.
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Andrus' total guarantee will be precisely $131.275 million over 10 years, including the $11.275 million he had left on his three-year deal through 2014. He would have been a free agent after '14.
The deal is expected to be announced sometime this week.
The extension might include an opt-out clause, MLB.com reported.
The Rangers had said they were going to make one last try at locking up Andrus, and had they been unable to lock him up may have been forced to trade him next winter. The Rangers now will be stacked at the middle-infield spots, with Ian Kinsler at second base plus their top prospect, shortstop Jurickson Profar, who was optioned to the minors late in spring.
The Rangers could wind up moving Kinsler to first base next year, or possibly elsewhere, as Profar is considered among the very best prospects in the game.
The talks were first reported by Ken Rosenthal of Foxsports.com.