The deal is for $9 million in each of the two years, the Los Angeles Times reported.
Street was thought to be able to fetch considerable value on the free agent market after the 2015 season, but Wednesday's signing could keep him in Los Angeles through the 2017 or even the '18 season. The deal contains a team option for a third year also at $9 million, according to the Times.
The 31-year-old had been slated to finish the final year of a six-season, $43 million contract he signed as a member of the Colorado Rockies.
Street came to the Angels from the Padres in July as part of a six-player trade that sent four prospects to San Diego.
In 14 appearances this season, Street is 2–1 with 11 saves and a 3.29 ERA. During his entire tenure with the Angels, he has only allowed one home run.
Street is tied with Rod Beck at 29th on the all-time major league saves list with 286 saves. He has the fifth-most saves of any active pitcher.