NRL.com said talks between the player and club were apparently progressing well, with a three-year offer on the table.
Harris made a highly successful switch from the edge to a middle forward last year, as the game sped up under the tweaks introduced by dynamic NRL boss Peter V'landys during the first lockdown.
The 29-year-old Harris "shapes as the likely captaincy replacement for Tuivasa-Sheck" from next year, according to NRL.com.
Meanwhile Tuivasa-Sheck, who is on about $1m a year, said it was "tough to tell the boys" about his move to rugby next year.
"It's my last time with the club and these players," Tuivasa-Sheck said.
"But I'm excited for it. There's nothing extra that I need to do or want to do.
"I've just got to go out there and put my best foot forward for the boys to follow."
The Warriors play the Titans in a trial at Lismore in north eastern NSW on Saturday. Under the new feeder club partnership, those missing the 24-man squad will play for Brisbane club Redcliffe this weekend.