The England and Gunners winger, 23, is in the final year of his current deal and could leave for free next summer.
He told SunSport last month that he wanted to stay at the Emirates Stadium.
But Arsenal chairman Peter Hill-Wood said: “We have got our general overall wage structure and we are not prepared to break it for anybody.
“We have made an offer which he didn’t think was acceptable and that is the end of it. We think it is a very reasonable offer.
“For the moment his people are still talking to us and we hope it will be resolved. But don’t ask me when, because I haven’t a clue.
“I hope that he will eventually come round and sign.
“It will be a pity if he does not but the whole issue of pay for footballers has got totally out of hand and they don’t get terribly well advised a lot of the time.”