Reprisals have eased only in recent years. The ban on the US exercising with New Zealand was lifted only in 2010.
US ambassador Mark Gilbert has said the US would not relax its long-running order to refuse to confirm or deny if a Navy ship was carrying nuclear weapons.
Mr Key this afternoon said that obstacle could be overcome, and he could accept the advice of Ministry of Foreign Affairs and Trade (Mfat).
"How they form that view, I don't know - whether they specifically ask them or whether they do it through open-source intelligence. There are a lot of ships that are neither nuclear armed or carry nuclear weapons."
Under New Zealand law, ships may visit only if the Prime Minister is satisfied they are not carrying nuclear weapons.
According to a new Newshub Reid Research poll, 75 per cent of respondents supported a US Navy ship visit. Twenty per cent were opposed, and 5 per cent did not know.
In January, the results of a Herald DigiPoll showed a total of 29.4 per cent didn't want a ship to visit at all; 50.2 per cent think it would be a positive move; and 16 per cent displayed a sense of triumphalism by preferring to think it would be a victory for New Zealand's anti-nuclear policy.