Siddons, the Wellington coach who has two years left on his contract, said the job piqued his interest but with a young family it wasn't the right time.
He was formerly Australia's batting coach under John Buchanan and has a proven track record as a batting specialist, with Bangladesh and the Firebirds.
Edgar, the national selection manager under coach Mike Hesson, was a qualified batting coach in Sydney before returning to Auckland. He also seemed a good fit, but the former New Zealand opener said he was happy in his current role.
That leaves McMillan, who stood in for Carter in the Wellington test against West Indies in December.
He tasted success as Canterbury's batting coach when they won the Plunket Shield. Just four centuries were achieved, although as many as fourteen players averaged over 20 with the bat.
Canterbury's head coach, Gary Stead, is understood to not be putting his name forward.
McMillan's strength is seen as taking batsmen out of their comfort zones and trying to bridge the gap between domestic and international standard. The drawback for the 55-test batsman is the amount of time away from his family.
New Zealand Cricket says its preference is to employ a New Zealander. Other contenders, if they apply, would be Northern Districts' James Pamment, Auckland's Matt Horne and former Knights coach Grant Brad burn, who also coached NZA. Wildcards would be Martin Crowe and Mark Greatbatch, although it's doubtful either would apply.
Applications close on May 6, and the team depart for the Caribbean on May 24.