Josh Morris, 33, was reported to have requested a move to the Roosters to play alongside his brother Brett.
Sharks coach John Morris confirmed the veteran was considering his options, but ruled out a move for the New South Wales representative.
"There was an enquiry there around Josh's side and I think our fans and members deserve the full truth," he told the NRL website.
"There is an option there for him to take up a deal to play with his brother, but as head coach of the Sharks I have got to make sure I put out a competitive team on the park every week that can win us games and hopefully win the grand final.
"Josh Morris is our most experienced player and he will be here in 2020. We have just lost 1,000 games of experience with Paul Gallen gone, Matt Prior gone, Sosaia Feki gone, Kurt Capewell gone and Jayden Brailey gone.
"Apart from 'Gal', a lot of these are the results of a pretty horrific salary cap fine that was levelled against our club and that happened before I started, but I will do my best to bring the club through it."
Formerly of the St George Illawarra Dragons and Canterbury Bulldogs, Josh Morris arrived at the Sharks for the 2019 season.
The centre scored 15 tries as Cronulla finished seventh before falling in an Elimination Final to the Manly Sea Eagles.
But Josh Morris is reportedly unhappy at the Sharks with a move to the wing likely in 2020 after Jesse Ramien's return to the club from the Newcastle Knights.