While those talks continue, Rose also met with the Los Angeles Lakers on Thursday, sources told ESPN's Chris Haynes and Ramona Shelburne. Sources told Haynes the meeting with the Lakers lasted three hours.
The former No. 1 draft pick has met with several teams over the past three weeks, including the Milwaukee Bucks and LA Clippers, but he has gained momentum on a deal with the Cavs in the past several days, sources said.
Rose is expected to have a face-to-face meeting with the Cavaliers in the coming days, league sources told ESPN.
He is not expected to make a decision until next week.
Playing on a contender is important to Rose. If he agrees to a deal in Cleveland, it would also allow him to be closer to family in Chicago.
The Cavs are believed to be offering Rose a minimum contract for $2.1 million for the upcoming season to play with Kyrie Irving in the team's backcourt. Adding Rose in this manner would add $6.4 million to the team's luxury tax bill for next season, which would be a projected $78.2 million.
The Lakers are trying to entice Rose to sign with them, suggesting they can offer more playing time and money in a better environment after Rose's tumultuous 2016-17 season with the New York Knicks, sources said.
Los Angeles thinks Rose, who has been working out at the Lakers' practice facility lately, can help the development of rookie point guard Lonzo Ball.
The Knicks have had minimal contact with Rose during the free-agency period, league sources told ESPN's Ian Begley.
In 2011, Rose became the youngest player in NBA history to win the MVP award, as a member of the Chicago Bulls.
He averaged 18 points, 3.8 rebounds and 4.4 assists on 47.1 percent shooting -- his best percentage since the 2009-10 season -- in 64 games for the Knicks last season.
Rose also averaged a career-high 10.1 points in the paint last season, which tied for second among NBA guards behind Toronto's DeMar DeRozan (10.3).
ESPN's Bobby Marks contributed to this report.