Nash signed with the Lakers last week and he tweeted word of Hill's signing with the other team in Staples Center earlier this week via a photo of him frowning on Twitter.
Hill signed a two-year deal on Wednesday. He turns 40 in October and is expected to back up starter Caron Butler at small forward after reportedly choosing the Clippers over the Lakers, NBA champion Miami Heat and the New York Knicks.
He played the past five seasons in Phoenix, averaging 10.2 points, 3.5 rebounds and 2.2 assists in 28.1 minutes while playing in 49 games last season. Before moving west, Hill spent seven injury-filled seasons in Orlando, where he was limited to 200 games because of left ankle problems.
Hill, an 18-year NBA veteran, received votes for the All-NBA defensive team and Defensive Player of the Year last season.