Ramsey is out of contract in the summer and Bayern, along with other interested parties, are able to speak to him now with a view to agreeing a deal but the Wales international arriving sooner than that looks unlikely.
Paris Saint-Germain, Real Madrid and Juventus have also approached Ramsey, who scored in the 4-1 win over Fulham yesterday, but Gunners boss Unai Emery does not expect him to try and force a move away this month.
"At the moment he is working with us and I think yes [he will stay until the end of the season]," the Spaniard told reporters. "He worked very well in Liverpool.
"Today [Tuesday] he played 15 minutes and he scored. He helped us for confidence and to earn a good result. I want from him this focus when he is playing."
Bayern are looking to revamp their squad as Franck Ribery and Arjen Robben head for the exit door this summer and Ramsey has emerged as a top target.