Jose Mourinho is said to be interested in bringing Arnautovic to Old Trafford, perhaps as early as January, in order to bolster his options in attack.
RB Leipzig's Sabitzer thinks his international team-mate is ready for a move to United after showing his "real class" in recent seasons.
"Marko is a great guy," Sabitzer told Omnisport. "I get along well with him and we have played together for a long time with the national team. We are good friends and once went on holiday together. I like him a lot and I think he feels the same way about me.
"Everybody can see that he has athletic qualities. He had a lack of maturity and luck in Germany and I think he knows that. But his performances in England the past few years have demonstrated his real class and I can see the same in terms of the national team.
"That's why I believe that he can make another step forward and play a major role there [at Manchester United]. I cross my fingers for him."
Arnautovic, who is sidelined with a hamstring injury and is not expected to be fit until next month, has scored five goals in 13 Premier League appearances this season.