The 6-foot Lowry averaged 9.1 points, 4.5 assists and 3.6 rebounds per game last season backing up Aaron Brooks.
"He's a big part of our team," Morey said in a phone interview. "He did a great job off the bench and we also like he and Aaron playing together. That's very big for us to get him back."
Lowry's agent, Andy Miller, did not return multiple messages seeking comment. The Rockets had seven days to match Cleveland's offer, but Morey saw no reason to wait.
"We like to let the player know as soon as possible, just so he doesn't start thinking about playing for another team," Morey said.
Scrambling to bolster their roster since MVP LeBron James announced he was leaving last week, the Cavs have come up short so far this summer. On Tuesday, 14-year veteran Zydrunas Ilgauskas said he will follow James and sign with the Miami Heat.
James' decision last week not to re-sign with Cleveland and bolt after seven seasons has put the Cavs in a bind. Most of the other top free agents have signed elsewhere, leaving the team with little on the market to choose from. Cleveland still has a $14 million trade exception to use and the club has had talks with several teams about deals, but nothing is imminent.
Lowry and starting forward Luis Scola were Houston's restricted free agents, and Morey said re-signing Scola is the next priority.
"Hopefully, that's not too far off," Morey said. "We're going to bring him back, that's been the plan all along."