The deal confirmed by the Clippers hours after the NBA trade deadline passed reunites Green with Doc Rivers, who coached him in Boston from 2011-13.
The 29-year-old forward averaged 12.2 points, 4.5 rebounds and 1.8 assists in starting 31 of 53 games for Memphis this season. The seven-year veteran has also played for Seattle during his career.
Stephenson averaged 4.7 points, 2.5 rebounds and 1.4 assists in starting 10 of 43 games for the Clippers. Rivers said that the 25-year-old guard had "a good impact" on the team during his short time in Los Angeles. Stephenson has also played for Charlotte and Indiana in his career.
ESPN.com first reported the deal.
The Grizzlies traded for Green in January 2014 as part of a five-player trade with Boston and New Orleans. His contract expires after this season.
With centre Marc Gasol out indefinitely with a broken bone in his right foot, this is the second trade the Grizzlies have made since Tuesday rearranging the roster for a team that currently sits fifth in the Western Conference with a 31-22 record having made five straight playoff appearances.
The Grizzlies also traded guard Courtney Lee to Charlotte on Tuesday as part of a three-team trade with Miami that brought centre Chris Andersen and forward P.J. Hairston to Memphis along with four second-round draft picks.