A current best estimate, based upon genetic data, places the size of the founding population of the Americas in the range of 229 to 300 with a best fit of about 284 people.
The source scientific journal article for the material in the link is as follows:
In spite of many genetic studies that contributed for a deep knowledge about the peopling of the Americas, no consensus has emerged about important parameters such as the effective size of the Native Americans founder population. Previous estimates based on genomic datasets may have been biased by the use of admixed individuals from Latino populations, while other recent studies using samples from Native American individuals relied on approximated analytical approaches.
In this study we use resequencing data for nine independent regions in a set of Native American and Siberian individuals and a full-likelihood approach based on isolation-with-migration scenarios accounting for recent flow between Asian and Native American populations. Our results suggest that, in agreement with previous studies, the effective size of the Native American population was small, most likely in the order of a few hundred individuals, with point estimates close to 250 individuals, even though credible intervals include a number as large as ~4,000 individuals.
Recognizing the size of the genetic bottleneck during the peopling of the Americas is important for determining the extent of genetic markers needed to characterize Native American populations in genome-wide studies and to evaluate the adaptive potential of genetic variants in this population.
Nelson J.R. Fagundes, et al., "How strong was the bottleneck associated to the peopling of the Americas? New insights from multilocus sequence data", 41(1) Genetics and Molecular Biology (2018).
This number is the "effective population size" of the Founding population of the Americas which is generally significantly smaller than the actual adult census size of the same population, and requires additional adjustment for non-reproductive age adults. Depending upon the circumstances, effective population size could be somewhat more than half to less than 1% of the total census size of the population.
The review of the literature in the paper recounts many previous estimates of the same quantity, after which the authors argue that their approach is better than their predecessor's approaches, despite a quite small data set that was analyzed.
The first quantitative approach to infer the effective population size of the founder Native American population was developed by Hey (2005), who did a meta-analysis of nine sequence loci, used a likelihood-based inference and assumed a isolation with migration (IM) population model to suggest an extreme population bottleneck with an effective population size of ~70 individuals. Since this pioneer work, other groups tried to replicate this result using multilocus autosomal data, with partial success. Kitchen et al. (2008) re-analyzed Hey’s dataset, adding mtDNA genomic data under different priors for migration rates and suggested an effective population size ranging from 1,000 to 5,400 individuals. Ray et al. (2010), using a dataset of 401 STRs, estimated an effective founder population size between 42 and 140 individuals (with a median of 87 individuals). Between these two extremes, Fagundes et al. (2007), based on the re-sequencing of 50 short loci, estimated an effective founder size of ~450 individuals (with a 95% credible interval (CI) ranging from 71 to 1,280 individuals). Recent autosomal data generated from admixed Latino populations also provided very different figures. Gutenkunst et al. (2009), based on a very large dataset of more than 13,000 SNPs, suggested a value of 800 effective individuals, with a confidence interval between 140 and 1,600 individuals; while Wall et al. (2011), using resequencing data, estimated a bottleneck effective population size not larger than 150 individuals. Gravel et al. (2013) proposed intermediate values of about 514 effective individuals, ranging between 316 and 2,264 individuals.