If it's a personal union for Iceland then the King could be pressured to sell or transfer ownership to the USA.
It's the 20th Century, by this point it's well established that rule is with a constitutional monarch, and exercised by the Althingi - Icelandic Parliament as the successor to the Danish Parliament. I doubt that the Danish King could hand over the reigns even if he wanted to. Legally speaking (as a complete legal non-expert!), the treaty establishing the Kingdom of Iceland placed it in a personal union with Denmark until 1940, at which point the personal union could be renegotiated, or scrapped with 3 years notice (as they did in OTL). Such a theoretical transfer of sovereignty would run against that treaty and domestic laws implementing it's provisions.
Realistically, there is no way the Icelanders would agree to such a transfer. They, as a clearly identified people and cultural group, spent virtually a millenium under Norwegian or Danish rule, and have just regained full independence and sovereignty, with Denmark supporting them. No way they'll consent, even passively, to just handing it away to a some foreigners who have no affinity for Nordic, let alone Icelandic, culture. If such a thing was attempted, you can be certain there'll be appeals to the Danish Supreme Court to deal with it, an attempt to dispose of the King and establish a Republic earlier (as many citizens wanted in their hearts in 1918, but didn't want to bite off more then they could chew in one go) in Iceland, and absolutely no desire to accommodate any kind of western/NATO military base in Iceland in future (so no Keflavik Air Base for the US).
Edit: Just to add to this, here are the relevant passages of the 1920 Icelandic Constitution (see below). The King can not amend who inherits the crown without the approval of the Icelandic Parliament, he must swear an oath to the Icelandic Constitution, his powers are supreme but limited by the Constitution, and
he can not make any foreign agreements which make obligations of territory or territorial waters without the consent of Parliament.
Article 3
The decree of royal inheritance is that which is stated in Articles 1 and 2. royal succession law of July 31, 1853. Royal inheritances may not be changed, except with the approval of Parliament.
Article 6
The king swears an oath to the constitution when he comes to the state, provided he has not worked on that oath while he was a state heir. Of this oath, two identical originals shall be made. The Althingi will stores one, and the National Archives collects the other.
Article 9
The king has supreme power in all affairs of the state, with the restrictions, which is set out in this Constitution, and he has the Minister to implement it. Ministry is based in Reykjavik.
Article 17
The king makes agreements with other states. However, he can make no such agreements, if they imply surrender or obligations on land or territorial waters, or if they look to change them unless the approval of the Althing is passed.