Xhaka seemed set to leave Arsenal in this month's transfer window, having been heavily linked with a move to Hertha Berlin.
The Switzerland international was replaced as club captain by Pierre-Emerick Aubameyang after a confrontation with Arsenal fans during a home match against Crystal Palace in October.
Former Borussia Monchengladbach midfielder Xhaka had reportedly agreed terms with Hertha, but Arteta now believes the 27 year-old will remain at Arsenal.
"The first message for him was that I wanted to understand how he was feeling, why he was feeling that way," Arteta, who has won two of his four matches in charge of the Gunners, told reporters.
"I wanted him to hear from me my opinion on him, and that I was ready to support him and that I was ready to push the club to be supportive of him too, because I thought he could be a really, really good player for us and he could enjoy in this football club playing under me.
"I tried to convince him that way. He thought about it, he had a very positive response afterwards and I think he changed his mind."
Arteta is also unconcerned over the aftermath of the incident against Palace and believes Xhaka's relationship with Arsenal's supporters will soon be on the mend, despite his previous desire to leave the club.
"It is very common now. Every club has its own issues. Some things within the club, some within the media, some from social media," Arteta added.
"It is very difficult nowadays to have one full year where everything is perfect.
"It is the environment, we have to deal with that and hopefully in a few months nobody remembers these issues. Of course it is not ideal when players respond and it creates an instability. So as much as we can avoid it, the better."