Persistent recruitment chatter has led the coach to call on the NRL to overhaul its "ordinary" transfer system after Tago became the latest Panther in the headlines.
A personal matter left him benched to begin the season and replacement Tom Jenkins' starring performances in the opening two rounds have clouded his path back to the starting side.
Three-time premiership winner Tago holds the equal-longest contract of any active Panther, tied to the heavyweight side through 2029.
But reports emerged at the weekend Penrith could engage in a swap deal for disgruntled Bulldogs centre Bronson Xerri given Tago is struggling for game time.
"Someone's lying. We love him," Cleary said of Tago.
"I know it's good for you guys (the media) because you get to run with it but someone there is telling porky pies.
"It's part of the reason our transfer system, if you want to call it, I think is an ordinary one, where it's just 365 days a year."
Cleary would prefer a system where trade was confined to an allotted window in the off-season.
Such a system runs successfully in the AFL, though would unlikely to be supported by the Rugby League Players Association given players would have less time to organise future plans.
"We've certainly seen in other sports it (a transfer window) adds to that period where there's build-up to it and fans can get involved in it," Cleary said.
"The way we run now, on a daily basis, we're almost having recruitment meetings. It's crazy."
Tago was among a handful of players hitting the training paddock at Penrith's centre of excellence early on Monday morning.
He was joined by star second-rower Liam Martin, who could return from a calf injury for Friday's round-three clash with the Sydney Roosters.
"He's over the injury, we're just looking to get a few more k's in the legs. He's definitely a good chance," Cleary said.
Tago has played one game of NSW Cup this year and featured on Penrith's newly-extended bench for their start to 2026, though went unused in wins over Brisbane and Cronulla.
"We've got all the faith in the world in 'Tags'. He's kept his head up and trained really hard and each week, he's always a chance to get on the field anyway," Cleary said.
"It's a long, hard year and we're going to need him. You don't know when that's going to be."
Perth-bound prop Liam Henry is unlikely to recover from his knee issue in time to face the Roosters.
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