He will replace Geno Smith, who is also a free agent, as the Chargers' backup quarterback behind starter Philip Rivers.
Taylor, 29, was acquired by the Cleveland Browns in a trade with the Buffalo Bills last year. The expectation was that he would be the team's starting quarterback for the entire season while a young quarterback selected in the draft would sit and learn behind Taylor.
Taylor, however, left the third game of the season against the Jets with a concussion, and Baker Mayfield took over and guided the Browns to their first win since Christmas Eve of 2017.
Mayfield never let go of the job, while Taylor threw just one pass the rest of the season -- an 11-yard completion on Nov. 4 while Mayfield was being checked for a concussion.
In three games, Taylor was sacked 13 times and went 42-for-85 for 473 yards with two touchdowns and two interceptions (64.5 rating). He also earned $16 million -- $6 million in a roster bonus -- and was the Browns' most expensive player under the salary cap in 2018.
Taylor played better and longer in Buffalo, where he started 43 games in three seasons and completed 62.6 percent of his passes with 51 touchdowns and just 16 interceptions. He was the Bills' career leader in passer rating (92.5), and his 1,575 rushing yards are also the most in franchise history.
ESPN's Pat McManamon contributed to this report.