Taylor could be back as soon as Sunday for the Chargers' game against the Jacksonville Jaguars as Justin Herbert's backup, Rapoport adds.
The veteran quarterback has been out since Week 2 after a team doctor accidentally punctured his lung while trying to administer a pain-killing injection to his cracked ribs.
Taylor came into the season as the team's starting quarterback but was overtaken by Herbert after the rookie played impressively well during his absence.
Herbert has thrown nine touchdowns and three interceptions in four games this season.
Taylor completed 53.3% of his passes for 208 yards in his only start of the season, a Week 1 victory over the Cincinnati Bengals.