The midfielder has spent the last decade with the Gunners and had hoped to remain at the Emirates Stadium, but the North London outfit have opted to withdraw the offer of a new deal.
Ramsey has been heavily linked with a move to another top European club in January, but the playmaker has revealed that he wants to try to "achieve something special" with Arsenal during the rest of the season.
The 27-year-old is quoted by The Telegraph as saying: "I am contracted to Arsenal and I am going to do my best this season to achieve something special.
On his contract, Ramsey added: "That's a decision that they have made - things happen in football and you just have to get on with it and carry on playing the best you can," he said.
"Everything has been going great with the club. We thought we were in a position where we had agreed a deal but that's no longer the case. So I just have to carry on playing my football and do my best for Arsenal this season and I'll leave the rest with the club now."
Ramsey has contributed one goal and three assists in eight Premier League outings this season.