Vettori, who injured the ankle during training on Thursday, has already been ruled out of the domestic Twenty20 final on Sunday, New Zealand Cricket (NZC) said on Friday.
Off-spinner Jeetan Patel has been called into the squad as cover should Vettori be ruled out of the first match in Auckland next Tuesday.
"He (Vettori) will join the Black Caps when they assemble in Auckland, where he will be assessed by medical staff," team manager Lindsay Crocker said in a NZC statement.
"A decision on his availability will be made next week, ahead of the opening game of the England tour at Eden Park. Brendon McCullum will captain the team should Daniel not be available."
New Zealand meet England in Twenty20 matches on Feb. 5 in Auckland and in Christchurch on Feb. 7, before playing five one-day internationals and a three-Test series.