(I consider the FIDE rules to be the "the rules of chess" and this answer is only about them)
No, you cannot claim. In fact your right to claim ends as soon as you touch a piece.
Here's what the rules say, emphasis in bold mine:
9.2.1 The game is drawn, upon a correct claim by a player having the move, when the same position for at least the third time (not
necessarily by a repetition of moves):
9.2.1.1 is about to appear, if he first writes his move, which cannot be changed, on his scoresheet and declares to the arbiter his
intention to make this move, or
9.2.1.2 has just appeared, and the player claiming the draw has the move.
So either it is about to appear, or it has just appeared, and not at any other time.
Then 9.4 ends the debate:
9.4 If the player touches a piece as in Article 4.3, he loses the right to claim a draw under Article 9.2 or 9.3 on that move.
That's related to the touch move rule -- once you touch a piece that can legally move with the intention to move it, you have to do so. It's too late for claims.
If it were possible to claim at any later time, there would never be any reason to continue: as soon as it was clear one of the players was going to lose, they'd claim the draw.
But in the case of five fold repetition, it's different:
9.6 If one or both of the following occur(s) then the game is drawn:
9.6.1 the same position has appeared, as in 9.2.2 at least five times.
If a position repeats five times, then that's just an automatic draw, no claim needed. If it is found out later during the "game" that a position was repeated five times some time ago, then that means that you just weren't playing the game anymore after that as it was already finished. You were moving pieces about.
