Xhaka had reportedly agreed to join Hertha Berlin in the transfer window, with his future at Arsenal in doubt ever since he reacted angrily to being replaced during the 2-2 draw with Crystal Palace in October.
But after missing Arsenal's 2-1 defeat to Chelsea on December 29, Xhaka started 2020 in positive fashion with a strong performance in his side's 2-0 victory over Manchester United on New Year's Day.
Asked whether the 27-year-old had given him assurances that he would stay at Emirates Stadium, Arteta said: "He did, yes.
"I think he's going to stay. I really like him. If we get him on board, he can be a tremendous player for us."
First-half goals from Nicolas Pepe and Sokratis Papastathopoulos helped Arteta to his first victory since taking over as Arsenal boss and he praised his side's improved performance.
Arteta suggested there is more to come from his players, who he said are not yet at the level of fitness he expects of them.
"Everything we tried to do against them worked really, really well," Arteta said.
"I think we had enough chances to put the game to bed much earlier and in the second half we knew that physically we were going to drop: at the moment we are not ready to maintain that level of intensity through 90 minutes.
"But the spirit, the commitment and the desire the boys are playing with at the moment is terrific."
Gunners defender David Luiz echoed his coach's comments, and indicated that Arteta's arrival has transformed the spirit in the Arsenal camp.
"We have to be honest and humble to admit: physically we are not ready," said Luiz. "We showed that in the second half against Chelsea, and we showed that today.
"This team is in a moment now to change a lot of things. We cannot change from zero to 10 in one day, one week or one month.
"Mikel is a great coach. He knows football, he was a great player. I believe in his philosophy and he can improve every single player and in life when you're happy the results can be totally different.
"If you sleep happy you can sleep four hours and it's better than sleeping sad for eight hours. If you work with happiness, believing what you are doing, it's totally different."