There have been a number of genetic studies comparing modern dog populations with archaeological remains of dogs. However, even before these studies the conventional archaeological wisdom was that the Paleoindian people who populated the Americas brought dogs with them from eastern Asia. The reasoning was that the earliest domesticated dog remains in eastern Asia far predated the peopling of the Americas, and so whichever populations moved into the Americas likely already had dogs.

Genetic testing has more or less confirmed this suspicion, that east Asian dog populations are closely related to the earliest dogs found in the Americas. For instance, the oldest dog remains in the Americas (from the Koster site in Illinois) are genetically quite similar to a number of dog remains found at Zhokov Island in the arctic, off the coast of eastern Siberia (Leathlobhair et al. 2018). In the broader story of dog domestication, both this study and a number of earlier studies (including most recently Frantz et al. 2016) indicate that dogs fall broadly into two groups, suggesting independent domestication in two regions of the Old World. From Leathlobhair et al. 2018:

Studies of nuclear genome data have identified two modern clades of global dogs: an East Asian clade (including dingoes) and a Western Eurasian clade (including European, Indian, and African dogs)

The exact relationships between these groups is unclear, and there may have been significant intermixing of the east and west Asian dog populations due to human movements (for instance, movements of people across central Asia introducing east Asian dog genes into west Asian dog populations). Likewise, as demonstrated by the Koster case and testing of a number of other archaeological dog remains, New World dogs prior to European contact cluster most closely with this East Asian clade.

However, there was likely pretty significant divergence between this East Asian clade and the dog populations that ended up in the Americas, due to the isolation of these two populations following the original peopling of the Americas. There may have been a second introduction of dogs into the New World by paleo-Inuit people more recently (the last few thousand years), and archaeological remains of Arctic dogs in Alaska from this more recent period are quite similar genetically to Arctic dogs from Siberia. Likewise, while the people of the Americas did not domesticate dogs, there was certainly continued manipulation of dog genetics in the Americas, including the possible introduction of some coyote DNA into American dog breeds (see Leathlobhair et al. 2018).

Finally, these genetic studies have also demonstrated that modern dog breeds in the Americas are largely descendants of the West Asian clade of dogs. The introduction of West Asian dogs in the last 500 years has largely replaced indigenous dog populations, such that even breeds which are known archaeologically as New World breeds (like chihuahuas) are in the present actually from this West Asian stock. As such, although modern chihuahuas morphologically resemble pre-contact chihuahuas, they are not genetically related. Most likely, these pre-contact breeds were replicated by breeding Eurasian stock following the replacement of pre-contact dog breeds with Eurasian dogs. The only possible exceptions are some arctic breeds (like malamutes) and Carolina dogs, though in both cases there isn't a clear-cut relationship with pre-contact dog breeds (probably because of intermixing with West Asian dogs).

Sources:

Frantz, Lauren A. F. 2016. Genomic and Archaeological Evidence Suggest a Duel Origin of Domestic Dogs. Science, 352(6290):1228-1231.

Leahtlobhair, Maire Ni, et al. 2018. The Evolutionary History of Dogs in the Americas. Science, 361(6397):81-85.