I have copied this from Wikipedia and I fear that it will turn out to be more than just a rumour.
Commonwealth relations saw renewed interest during the second term of United States president Donald Trump. In 2025, during Russia's invasion of Ukraine, Commonwealth countries Australia, Canada and New Zealand aligned themselves with the "coalition of the willing" spearheaded by British prime minister Keir Starmer to provide support for Ukraine.[58] In Canada, during the 2025 Liberal Party leadership contest, and in light of aggressive actions taken by the United States, the ultimate victor Mark Carney expressed support for deepening relations with Britain, Australia and New Zealand.[59] Donald Trump expressed interest in the United States becoming a member or associate member.[60]
I don't know if there is any consultation with Commonwealth members about who is allowed to join or if it is in the King's gift as Head. My thoughts on it are that as the position of Head of the Commonwealth is not hereditary, if America is invited to join then Trump will immediately want a deal that ousts the King and puts Trump in charge.