Billups, who currently works as an analyst for ESPN, has had a relationship with Gilbert for more than a decade. Billups endorsed Rock Financial, Gilbert's Detroit-based mortgage company, when the former NBA point guard was a member of the Pistons.
News of Cleveland's interest in Billups came on the same day that the Cavaliers parted ways with GM David Griffin. Griffin's contract was up at the end of the month, but negotiations between the two sides broke down.
In addition to his relationship with Gilbert, Billups is also close with Cavs head coach Tyronn Lue. The two dined together several times during the NBA Finals.
ESPN's Brian Windhorst contributed to this report.