The clubs announced the trade Wednesday, four days before the Chargers' first full practice of training camp.
The 6-foot-5 Jones led Ohio State to a national title during the 2014 season. After one more year with the Buckeyes, he was a fourth-round pick by Buffalo in 2016.
Jones was a project for the Bills, where Chargers head coach Anthony Lynn spent last season as running backs coach before his promotion to offensive coordinator. Jones was inactive for 15 games in Buffalo before throwing 11 passes in the fourth quarter of the season finale.
His future in Buffalo was in question after the Bills drafted Pitt quarterback Nathan Peterman in the fifth round last April.
The Chargers are looking for a mobile, athletic third-string quarterback behind Philip Rivers and Kellen Clemens, who has been Rivers' backup for three seasons. Undrafted quarterbacks Mike Bercovici and Eli Jenkins also are on the roster.
Robert Griffin III worked out for the Chargers on Tuesday, but wasn't signed.
Rivers is a six-time Pro Bowler who has started 185 consecutive games.
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