Matthews would likely get a minor league contract. He hit .190 with one RBI in 58 at-bats with the Mets.
Cincinnati would be Matthews' eighth team. Reds manager Dusty Baker is close friends with Matthews' father, who also was an All-Star outfielder.
Baker did not address the report following Cincinnati's 6-4 win over the Oakland Athletics on Monday night. A team spokesman said only that Matthews had not yet signed with the club.
Matthews was an All-Star in 2006 with Texas. The switch-hitting center fielder signed with the Angels before the 2007 season, and was traded to the Mets for pitcher Brian Stokes last season. The Angels agreed to pay $21.5 million of the $23.5 million remaining on his $50 million, five-year contract.
The Mets acquired Matthews while waiting for injured All-Star center fielder Carlos Beltran to return. Matthews started on Opening Day and got two hits for the Mets, but Angel Pagan wound up getting the majority of playing time.
The Reds have been a surprise this season, and went into Monday night's game at Oakland in second place in the NL Central.