The 25-year-old suffered the problem on the opening day of the season during the Gunners' 4-3 defeat to Liverpool at the Emirates.
After his exploits at Euro 2016 for Wales, Ramsey claimed that he was unlikely to start the season, but Wenger included him in the starting lineup against Liverpool.
It was initially reported that the midfielder would be sidelined for around three weeks, but he has felt pain every time that he has attempted a training session.
"Aaron is still struggling with his hamstring and I don't think that he will be available against Basel or at Burnley so, before the break, he will not be available," Wenger told reporters. "We had a few disappointments because he was close to joining again and, every time, he is not completely ready. He is not far, but we have to be cautious with him.
"He has a small grade-one hamstring but he is not completely free in his mind yet to push completely, so we stop him every time when he has a little pain. He is in the last days of his rehab and, as long as he is not completely comfortable, we do not push him.
"I am a bit more cautious than I would be. If he was ready to play, I would involve him in the squad, but he is not. Will he go away with Wales? I don't think so."
Meanwhile, Arsenal are preparing for Saturday's clash against Chelsea at the Emirates.