Lowry spent last season with Houston, where he averaged 9.1 points and 4.5 assists in 68 games. The Rockets have seven days to match Cleveland's offer, which a person familiar with the contract said is for three years and includes a fourth-year option. The person spoke to the Associated Press on condition of anonymity because the Rockets have not yet indicated if they will try to retain Lowry.
The Cavs sent out a release confirming their offer to Lowry on Wednesday.
Rockets general manager Daryl Morey has said the team would match any offers for Lowry and starting forward Luis Scola.
Lowry's agent, Andy Miller, did not immediately return a message seeking comment.
James' decision last week not to re-sign with Cleveland put the Cavs in a bind. Because many of the top free agents were already off the board, they had to look at other ways of bolstering their roster.
The 6-foot Lowry, who was drafted by Memphis in 2006, is a proven defender. He could back up starter Mo Williams or perhaps move into the starting lineup if the Cavs make another move this summer.