New Zealand's general manager of selection Bruce Edgar revealed on Sunday that captain Brendon McCullum will return to the middle order for the Boxing Day Test, after opening throughout his country's recent series against Pakistan in the United Arab Emirates (UAE).
With McCullum replacing Anderson lower down, it appears Rutherford will step into the home side's XI alongside Tom Latham at the top of the order for the first of two Tests, as the 25-year-old is the only other player in the squad with opening experience.
Rutherford averages 27.08 over 25 innings in Tests with a top score of 171.
He last played a Test for New Zealand in June, making four and 19 in their third Test away to the West Indies.
McCullum averaged 69.40 as an opening batsman in the UAE, including a double century in the third Test, which New Zealand won by an innings and 80 runs to tie the series 1-1.
"This is another series, in different conditions and against a completely different opponent," Edgar said in a statement on blackcaps.co.nz.
"In these circumstances, we feel Brendon's better suited to, and offers more value, down the order."
Anderson injured his groin in New Zealand's fourth one-day international against Pakistan on Wednesday.
The 24-year-old has been replaced in New Zealand's 13-man squad by Dean Brownlie, who has not played a Test since May 2013.
Brownlie averages 29.62 with the bat in five-day cricket.
New Zealand Test squad: Brendon McCullum, Trent Boult, Doug Bracewell, Dean Brownlie, Mark Craig, Tom Latham, James Neesham, Hamish Rutherford, Tim Southee, Ross Taylor, Neil Wagner, BJ Watling, Kane Williamson