Rapoport said many of the team personnel he's spoken with this week at the NFL combine believe Rivers will ultimately sign with the Colts.
Additionally, the Las Vegas Raiders shouldn't be ruled out as a potential landing spot for the veteran passer, Rapoport added. The Raiders are also reportedly interest in signing Tom Brady.
The 38-year-old Rivers threw 20 interceptions last season - tied for the second-highest mark of his career - and his 23 touchdown passes were the fewest he's tossed since 2007.
However, Rivers would be playing behind one of the top offensive lines of his career if he joins Indianapolis.
It remains to be seen whether the Colts are interested in a short-term fix under center or if they'll look to the draft instead. Frank Reich's squad currently owns the 13th overall pick and is expected to do its due diligence on the top group of quarterbacks.