Italy international Zaniolo bagged the winning goal in the Giallorossi's Europa Conference League final triumph over Feyenoord last month and is under contract at the Stadio Olimpico until 2024.
But the off-season has brought frequent reports of a potential move away from Roma.
Zaniolo has also spoken of being flattered by the interest of other clubs in comments that have rankled Roma fans, and Giannini feels the player did himself no favours.
"If you want to stay at Roma, which is not a small team, you can also answer that, okay, you are happy to please others but that the goal is to stay where you are," said Giannini, pointing out he had not made similar comments during his career.
"Instead, his phrases lend themselves to many interpretations.
"I remember that Totti a few days ago said that he had spoken to them, to give him advice on how to live the Roma shirt to the fullest. I do not know if it has been heard."
Still, Giannini believes the club should look to keep Zaniolo, adding: "If you ask me, I would make him a staple of Roma. For his potential, for his age, for what he has already shown, he is a talent to invest in.
"Let's not forget that you won the Conference League thanks to one of his goals. But it is difficult to judge if you do not know the internal dynamics."
But the 57 year-old – a Scudetto winner with Roma in 1982-1983 – acknowledges the suggestion the club has not offered Zaniolo a contract extension indicates it has made its mind up.
"If this is true, it means that even Roma are not convinced to keep him," Giannini said. "Maybe they all agree on the idea of separating. So amen, let's move on."