He dislocated and fractured his thumb while attempting to take a catch in the fourth and final one-day international against England on Friday.
"A decision on the senior pace bowler’s availability for the upcoming ICC Cricket World Cup in India will be made early next week once the results of the surgery are known," NZC said in a statement on Wednesday.
New Zealand play champions England in their World Cup opener on October 5 and head coach Gary Stead said Southee would be given "every chance" to play a part in their campaign.
"We've got our fingers crossed the surgery goes well for Tim," Stead said.
"He'll have some pins or screws inserted in his right thumb and, providing the procedure is a success, it will be a matter of ensuring Tim can tolerate the pain and manage the actual wound when returning to train and play."
New Zealand have sent a second-string squad to play a three-match ODI series in Bangladesh beginning on Thursday.
