The Austria international sustained the injury in training earlier this week.
However, he is likely to be involved at White Hart Lane if his protective cast is given the all-clear by referee Mike Jones.
Stoke will also be boosted by the returns of Peter Crouch and Phil Bardsley from suspension, while Marc Muniesa is fit again after an ankle problem.
However, Dionatan Teixeira will miss out after breaking his foot on the same day Arnautovic was injured.
Hughes told Stoke's official website: "We had a horrible day on Wednesday with the couple of breaks for Dionatan and Marko."
Expanding on Arnautovic's situation, Hughes added: "His hand is in a cast and he should be okay to be involved on Sunday assuming the referee gives the cast the all-clear, which he should do."
In a subsequent press conference, Hughes heaped praise on winger Victor Moses, who has made a positive impression after arriving on loan from Chelsea.
"We are delighted with what Victor Moses has been producing," said the Welshman.
"As we stand, Chelsea, Victor and ourselves are all winning.
"We're not yet at the point to talk about Victor's future but I must say that he's done very well. Competition for places has helped."
