Immigrants are undocumented if they have crossed the border illegally or overstayed their visas. There are a lot of places where I can find the total number of undocumented immigrants, but where can I find a study that provides the number of illegal border crossings versus visa overstays?
A 2017 study from the Center for Migration Studies estimates the proprotion of visa overstays was 42% as of 2014 (the latest year they had data for at the time):
In 2014, about 4.5 million US residents, or 42 percent of the total undocumented population, were overstays.
What's more, the trend since 2007 has been that the majority of new incoming illegal immigrants are from overstays versus illegal crossings (they refer to them as EWIs, or entries without inspection):
Before 2007, well over half of all undocumented arrivals were EWIs. In fact, as recently as 2005, EWI arrivals exceeded overstays by about 120,000. The number of EWI arrivals began a steep decline after 2005, falling from 340,000 in 2005 to 140,000 in 2013. Since the crossover in these trends in 2007, the total number of overstays has exceeded EWIs by about 600,000.
The percentage of overstays increased somewhat from 1995 to 2000, stayed steady until 2004, and then increased rapidly from 2005 to 2010 (Figure 3). This occurred because, as we have seen, after 2005 the number of EWIs fell rapidly while overstays remained at roughly the same level. Overstays reached 61 percent of the total in 2010 and continued to rise, reaching about two-thirds (66 percent) of the total in 2014.
So it's hard to tell from this what the current numbers are five years later, but unless the trend has reversed, the current number is likely to be higher than their estimate of 42%.
And of course, the numbers in this paper (or any other paper looking at this) are subject to some wiggle room, chiefly because the government is the main provider of numbers, and they simply don't track this stuff very well at all.
This answer is intended to complement and expand on user3163495's answer above. PLOS also published a critical review of the study referenced above . In this response they point out several assumptions the the prior authors made that are not supported by empirical data and consequentially make their claim of an average 22 million undocumented highly speculative.
In the abstract:
Using emigration rates from the binational Mexican Migration Project survey for the illegal border-crosser portion of the unauthorized population, we generate a 2000 unauthorized population estimate of 8.2 million—slightly below Pew and DHS’s estimates—without changing other assumptions in the model. We conclude that this new model’s estimates are highly sensitive to assumptions about emigration, and moreover, that the knowledge base about emigration in the unauthorized population during the 1990s is not well enough developed to support the model underlying their estimates.
NPR, AP News, NBC News, and CBS News have each published an article on this in January 2019 (NBC's article was from October 2018).
They generally place visa overstays at around 58-64% of the total number of unauthorized residence in the U.S., that 66% of the total number of immigrants who joined the population in 2014 were from visa overstays, and claim that visa overstays outnumber uninspected crossings for the seventh straight year (alternative claim: that they outnumber uninspected crossings each year since 2007).
Many of these articles seem to be sourced by the CMS report from 2014, which seems to be one of the most current, comprehensive reports on both types of immigration. This report seems to focus more on total immigrants living in the U.S., as opposed to how many come here per year.
This Politico article states that there were 396,579 illegal border crossings in fiscal year 2018, and that there are about 400,000 per year. The Guardian states that illegal border crossings have lately (since at least 2009) ranged from 340,252 to 556,041 per year.
Extrapolating from all of the above information, there would figure to be about 625,000 visa overstays per year.
AP News reports that there were 701,900 visa overstays in fiscal year 2017.