The Grizzlies haven't yet submitted a formal offer, as Rivers must first clear waivers before signing a new contract.
Four or five other teams are still interested in the combo guard, according to Charania.
The Suns acquired Rivers and Kelly Oubre Jr. from the Washington Wizards on Monday in exchange for Trevor Ariza. After the deal, they quickly decided to part ways with Rivers, which could allow him to join a team with playoff aspirations.
The Grizzlies had expected to receive Oubre as part of a three-team deal with the Suns and Wizards, but the trade talks broke down due to a miscommunication over the players involved.
Instead, Rivers will provide Memphis with depth at both guard positions if he signs, as he's capable of being a playmaker and an off-ball scorer.
The Grizzlies (16-14) are currently vying for a playoff spot after finishing with the second-worst record in the Association last season.