Captain Cook, Ian Bell, Graeme Swann, James Anderson and Jonathan Trott all missed the Twenty20 series against the Black Caps, which the tourists won 2-1 with a brutal 10-wicket success in the deciding match.
Cook said that the trio of Swann, Anderson and Trott could be a little "undercooked" as they haven't played for England since before Christmas.
"Belly and I played a couple of weeks ago, so the other guys are probably a little bit underdone," he said.
The first ODI begins at 1am UK time at Seddon Park, Hamilton.
England: Cook, Bell, Trott, Morgan, Root, Buttler, Patel, Broad, Swann, Finn, Anderson, Bairstow, Woakes, Tredwell, Harris
New Zealand: McCullum, Guptill, Williamson, Taylor, Watling, Munro, Franklin, McCullum, Boult, McClenaghan, Mills, Elliott, Ellis