The deal is pending a physical.
Lincecum spent nine seasons with the San Francisco Giants, but he has not pitched in the majors since last June. Yahoo! Sports reported that the Chicago White Sox and Giants were two of the last teams to not make the cut.
The Angels have the direst need for a starting pitcher. Ace Garrett Richards was recently lost to a torn ligament in his pitching elbow. Tyler Skaggs is recovering from Tommy John surgery and C.J. Wilson is on the comeback trail from having bone spurs removed from his pitching elbow. Left-hander Andrew Heaney is dealing with forearm and elbow issues.
The Angels were forced to trade one of their minor league players last week to acquire a fifth starter to complete the rotation, making it possible Lincecum could become a starter.
Longtime ace Jered Weaver has struggled since going 3-0 with a 3.86 ERA in his first four starts.
The Angels earned a weekend road sweep at Seattle and are in third place in the AL West, trailing leader Texas by 5 1/2 games. They open the four-game Freeway Series against the Dodgers in Los Angeles on Monday night.
Lincecum, a two-time Cy Young Award winner, underwent arthroscopic hip surgery Sept. 3. He didn't pitch after June 27 last year because of degenerative hips and went 7-4 with a 4.13 ERA in 15 starts. That's after he pitched two no-hitters in an 11-month span against San Diego, in July 2013 and June 2014.
Lincecum wrapped up a two-year, $35 million contract that paid him $18 million last year. He spent the offseason rehabbing in Arizona rather than his regular Seattle-area routine.
The 31-year-old right-hander auditioned in front of representatives for all of the MLB teams on May 6 in Scottsdale, Arizona.
"I'm happy," Lincecum said then. "I was able to throw strikes with my pitches, stay within myself, commanded all of my pitches. I only had a couple of misses, and they weren't out over the plate, so that's encouraging for me."
One of baseball's best pitchers in his prime, Lincecum won the National League Cy Young Award in 2008 and 2009 and made four All-Star Game appearances. He helped the Giants win three World Series titles in five years.
Lincecum earned his nickname "The Freak" by defying the stereotype of what a big league pitcher is supposed to be, generating massive amounts of torque with his 5-foot-11, 170-pound frame.
Information from The Associated Press was used in this report.