The Grizzlies announced the move Tuesday.
Arenas, who finished last season with the Orlando Magic, worked out for the Grizzlies on Monday.
The Magic had acquired Arenas in a trade with the Wizards in the middle of last season but elected to amnesty the mercurial veteran before the start of this season. Arenas averaged 8.0 points, 3.2 assists and 2.4 rebounds over 49 games in primarily a reserve role for Orlando.
The Grizzlies are looking to beef up their scoring as they seek a return to the postseason.
ESPN The Magazine's Ric Bucher and ESPNLosAngeles' Dave McMenamin contributed to this report.