The Gunners completed a blockbuster deal to sign the West Ham star in a move that was worth £105million.
The England international was fresh from winning the Europa Conference League with the Hammers and is a regular starter for England, meaning that a move to the title-chasing Gunners was a deserved next step.
Rice was handed his debut in Arsenal's Community Shield clash with Manchester City on Sunday and speaking ahead of the game, Manchester United legend Keane was full of praise for Rice but believes that the 24-year-old faces a challenge to live up to his price tag.
Speaking to ITV Sport, Keane said: "He's obviously going to be surrounded by better players with different demands if he's going to play higher up the pitch.
"I think he definitely has that quality in terms of adding more goals to him, he's got that physical strength and he can get in the box.
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"They've obviously still paid way too much for him. He's certainly not worth over £100million, Declan Rice, but he's a really good player."
Responding to Mark Pougatch's point that they said Trevor Francis wasn't worth £1million, Keane said: "You can say for lots of players, I agree, but we'll find out over the next year or two how good Declan is.
"He turns up every week, he's a big, strong boy - again, we talk about that physicality that Arsenal lacked in the last month or two [of last season].
"Has he got that top quality in terms of seeing a pass and getting nine or ten goals? We'll soon find out."