Soria will get $8 million guaranteed, according to a source. With the signing, the Rangers' 40-man roster is at 38 players.
The 28-year-old right-hander is a two-time All-Star who had spent his entire major league career, which began in 2007, with the Kansas City Royals. He missed last season after having Tommy John surgery on his right elbow in April and will not be available to start the 2013 season.
Soria has three seasons in which he reached at least 30 saves. Among active pitchers with at least 100 career saves, the highest save percentages belong to Rangers closer Joe Nathan (89.5), Yankees closer Mariano Rivera (89.3) and Soria (88.9).
Rangers general manager Jon Daniels said Soria is throwing off flat ground up to 105 feet and that an "aggressive projection" on a time frame for a return could be May.
Daniels said the physical on Tuesday went well and that Soria has put on "good weight during rehab."
"We're encouraging him to take it slow and get it right because when he's right, he's such an impactful piece in the pen," Daniels said.
Daniels said Soria turned down a chance to close with other teams, opting instead to set up for Nathan.
"That speaks volumes about him," Daniels said. "He wants to win."