Xhaka was in the firing line earlier in the season after he reacted furiously to fans booing him when he was substituted in a 2-2 draw at home to Crystal Palace in October.
The incident led to him being stripped of the captaincy and also saw him left out of the side by Arteta's predecessor Unai Emery for a month.
Xhaka has started six of Arsenal's last eight league games but his agent said last week that the 27-year-old was targeting a move to Bundesliga side Hertha Berlin.
Arteta, however, said the midfielder has told him he wants to remain in north London, also praising him for his performance against United, which was Arsenal's first win in five matches and took them up to 10th in the standings.
"I think he's going to stay. I really like him. If we get him on board he can be a tremendous player for the football club," the Spanish coach told a news conference.
"I love the way he is, the way he lives his profession. He is a great football player. Today he was excellent at cutting out passes and he made a lot of great passes. I like how focused he is, he's willing to learn."
Arteta endured a difficult home debut, with Arsenal conceding two late goals to lose 2-1 to Chelsea on Sunday, but he was pleased to get off the mark and praised his players for how they reacted against United.
"I am very pleased with the performance and even more with the result. I said to the players what you transmit to the fans is up to you," he said.
"I want to say thank you so much to the fans because of the energy they brought."