The identity of that player is currently unknown but Hay will be naming a squad of 23 next week.
"They are really quite strict on vaccinations and getting into the country and needing to be double-vaxxed," Hay said. "At this stage there is a player that hasn't had that, and so we won't be able to select them at this point in time.
"Everybody's got to make their own choice around that. But with any choice comes consequences - sometimes the consequences are positive, sometimes they are not so good."
The All Whites are also targeting two more fixtures in January ahead of March's Oceania World Cup qualifying tournament in Qatar, when Hay will be hopeful of having no restrictions on his selection options.
As with the games in the October window, Hay anticipates having to select a squad for next month's games without any New Zealand or Australia-based players.
"It's a tough one for the A-League players at the moment but you can already see things are starting to open up, which is positive. Hopefully this is the last window where we are restricted in who we can select but, as was showed by the games against Bahrain and Curacao, I have every faith in the squad we will be assembling."
The All Whites have played teams from the world's second-largest continent only twice in the last decade.
The most recent was in June 2018, when a young side lost 2-1 to Kenya in Mumbai, with Sarpreet Singh marking his first international start with a goal.
The other was a goalless clash with South Africa in Auckland in May 2014, overseen by caretaker coach Neil Emblen, notable as the last match before Anthony Hudson was appointed.