The last person I remember citing this was Ted Cruz. Not that I necessarily believe him, but someone probably did the research. I rebutted your statement at the time but can't go looking for it at the moment.
You didn't rebut my statement any more than you could rebut the statement that 25% is less than 50%. It is undisputed that the majority of illegal immigrants are from Mexico and that the vast majority of illegal immigrants are from Mexico or Central and South America (the
numbers were 56% and 80%, respectively in 2006). Assuming that US visas are granted to citizens of other countries at rates roughly proportional to the population, and, except for Mexican border crossing cards, that seems to be the case, then it is nonsensical to believe that anywhere near 50% of illegal immigrants are visa overstays.
This is the visa statistics page of the State Department. For, 2015, there were 10,891,745 non-immigrant visas issued, of which 1,479,109 were for Mexicans. Of course, that latter number includes 1,166,668 border crossing cards (BCC), which are supposedly issued only to Mexicans who have very strong ties to Mexico and whom the State Department has good reason to go back to Mexico. If that in fact is not the case, then I would consider that to be an illegal border crossing, not a visa overstay. Regardless, even with BCCs included (and I think it makes no sense to include them), Mexican visas represent 13.5% of visas, yet Mexicans represent 56% of illegal immigrants in the US. It is quite obvious that visa overstays cannot even come close to accounting for the number of illegal immigrants from Mexico, and therefore the number of illegal immigrants generally, unless people from Mexico overstay their visas, and some huge multiple times the number that other visitors do.
What figure do you put it at? With cites and everything?
Well, using the data I linked to, I see roughly 600,000 visas (excluding Mexican border crossing cards) issued to people from Mexico and Central America. This is out of 9,725,000 total (once again, excluding Mexican border crossing cards). So 6.1% of visas are issued to Mexican and Central Americans, but they account for well over half of the illegal immigrants - say 75%. I'll assume uniform visa overstays across nationalities, since I don't see any reason why that shouldn't be the case, and I've never heard anybody suggest otherwise. Then if X% of the illegal immigrants were visa overstays spread uniformly across the world, and 100% of the rest (which is almost certainly too high) were due to illegal border crossings by immigrants from Mexico and Central America, we would expect the percentage of illegal immigrants from Mexico and Central America to be 6.1%*X%+100%*(100%-X%) = 100% - 93.9%*X, which would have to match 75%. This gives
X% = 26.6%. Of course, this number could be much lower if illegal border crossings weren't exclusively being done by immigrants from Mexico and Central America. In any case, X% can't be anywhere near 50% unless for some reason Mexicans and Central Americans were vastly more likely to overstay their visas than any other visitors. In which case, of course, we should restrict the number of visas we grant to people from those countries.