The Gunners made the shock decision to withdraw their contract offer to the influential midfielder last month, ending months of negotiations.
That leaves Ramsey, 28, in the position where is able to run down his existing deal and leave the Emirates on a free transfer next summer.
Arsenal had been reluctant to meet the Wales star's £250,000-a-week wage demands and decided to look elsewhere for a new player.
But it also leaves the club in the position where they will have to find at least £20million to sign a ready-made replacement.
So the Gunners may decide it would be better to find a buyer in the January transfer window to at least bring some money into the coffers.
Yet Ramsey insisted last week he is in no rush to leave and his happy to play out the rest of the season before securing a lucrative new move in the summer.
And Wenger believes his former club have handed all control of the situation over to the player – even if Unai Emery had a change of heart and decided he wanted to keep him.
He said: "I don't really know what is happening at Arsenal. Sure, Aaron Ramsey is in a strong position now and he will use this position.
"But I don't know how far Arsenal will go in terms of giving him a huge contract.
"Look, we tried that before and it can still happen.
"These things were once my responsibility, but not any more. I am happy about that."
Juventus had been linked with a possible move for Ramsey in January if the Gunners were happy to sell, but the midfielder now appears more set on taking his choice of clubs next summer.
And Manchester United could be one of the clubs who try to snap him up, as they consider attempting to sign him for the third time in his career.
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