The simple answer is no. By the time the Commonwealth was created, after WW1, the Anglo-American relationship had developed its own unique characteristics.

Its worth noting that Commonwealth wasn't really planned, it was the product of a series of compromises as British colonies gained independance. Essentially the commonwealth continues the beneficial imperial links while allowing member states a level of sovereignty that would be impossible had they remained colonies. America, having been independant for a century was too far removed from the empire to have ever been part of, or even considered including in, the conversations that lead to the Commonwealth.