Difficult question to answer, but a number of historians and anthropologists have tried. Most estimates tend to think about North America as a whole (outside of the urban societies of Mexico) rather than just the modern-day United States, so that's what I've provided below. Estimates tend to fall into one of two schools: the "low counters" and the "high counters."

The Low Counters: James Mooney was one of the first to seriously estimate the pre-Contact population of North America. His figure was about 1.15 million people. He came to this number by looking at ethnographic surveys from across the continent. A.L. Kroeber proposed a slightly smaller number of 900k in 1939. Douglas Ubelaker, in the 1970s, revisited the Mooney method and arrived at a figure of about 2.2 million for North America. He revised that figure down to 1.9 million in the 1980s.

The High Counters: Henry Dobyns in the 1960s represented a dramatic shift away from Mooney and was one of the first of the "high counters." He used what's called the retrospective projection method to estimate population---that is, he took the North American indigenous population in the 1930s, for which he had a relatively accurate figure, and applied a 90 percent mortality rate backward to 1492. He arrived at a pre-Contact population of 12.25 million, which he later revised to 18 million in the 1980s. The problem with this method is that it used the mortality rate observed in the sixteenth century in Mexico and the Caribbean. This rate, however, was not necessarily applicable to the entire continent. William Denevan and Russell Thorton are also "high counters" and estimated the pre-Contact North American population to be 4.4 and 7.0 million, respectively. Neither of these figures should be considered particularly accurate, though. Denevan arbitrarily doubled Ubelaker's number because he thought it was too low. Thorton likewise made rather arbitrary adjustments to Ubelaker and Dobyns.

The issue of pre-Contact population figures is highly politicized. Low population figures imply that North America was open for colonization by European powers. In the words of Cole Harris, to low counters, it was "convenient to think that the native population had always been small and that those who remained would soon die off." High figures, on the other hand, highlight the devastating nature of European colonization. Thorton, for example, described the post-Contact demographic trend as a "holocaust."

In conclusion, we don't really know what the population of pre-Contact North America was. Personally, I tend to favour the more rigorous methods of Mooney and Ubelaker to the retrospective projection method employed by Dobyns. I think Mooney's number is probably too low, given what we now know about population densities in the Mississippi valley and the Pacific Northwest. Ubelaker's figure of a couple million is perhaps the most reliable estimate. Still, we need to carefully assess, through improved archaeological research, settlement patterns and ecological carrying capacities on a case-by-case basis to come up with a more accurate figure.

Sources: Harris, "Voices of Disaster"; Mooney, The Aboriginal Population of North America; Ubelaker, "North American Indian Population Size: Changing Perspectives"; Thorton, American Indian Holocaust and Survival; Dobyns, Their Numbers Become Thinned; Denevan, The Native Population of the Americas in 1492; my historiographical paper of disease and depopulation in the Pacific Northwest in the 18th century (I'm happy to send you a copy if you are interested in reading it).