The 26-year-old would back up starter Jared Goff, who helped lead the Rams to a 13-3 record and Super Bowl LII appearance last season.
Bortles nearly led his Jags team to a Super Bowl the year before, but like Goff, his season ended with a crushing postseason defeat to the New England Patriots.
A 10-6 season and 2017 AFC Championship Game appearance seemed to foreshadow more success for the Jags in 2018, but the bottom fell out on Jacksonville en route to a 5-11 campaign.
Bortles was benched twice for Cody Kessler as he finished just 29th in adjusted yards per pass attempt, according to Pro Football Reference. This week the Jags signed ex-Philadelphia Eagles quarterback Nick Foles to a four-year deal, effectively ending Bortles' tenure in Jacksonville.
The backup route is Bortles' likeliest option for 2019, and the Rams would instantly have one of the best in the league if they sign the ex-UCF star.
The starter path is unlikely next season, with 31 of 32 NFL teams either having their option in place or looking likely to select a blue-chip signal-caller in the 2019 draft.
The 32nd team is the Miami Dolphins.
The Fins traded starter Ryan Tannehill to the Tennessee Titans on Friday, leaving former sixth-round picks Jake Rudock and Luke Falk as the depth chart's remaining signal-callers. Neither has started a game on the professional level, so it's unlikely either player is leading the Dolphins in Week 1.
But Miami can also look to the draft, where it picks 13th overall. The issue is Oklahoma's Kyler Murray, Ohio State's Dwayne Haskins and Missouri's Drew Lock may all be gone by then.
The Dolphins could conceivably sign Bortles, who is arguably the best quarterback left on the free-agent wire. But if that doesn't work out, then the 6'5" veteran can do far worse than Los Angeles, a dynamite offensive team that finished second in scoring.
Bortles could continue his development under head coach and offensive guru Sean McVay, who has pulled off many right moves during his two-year tenure.
In due time, Bortles could get another shot down the road as a starter following a backup stint, much like the man who won a Super Bowl before replacing him in Jacksonville.