What you're referring to is classed as EB5 immigrants. It's been around since the 1990s and for fairly good reasons, up until recently, few people actually took advantage of it. What changed was the introduction of a group of people who would gladly pay a lot of money even after understanding the downsides.

Typically, today what happens is that you sign up with a regional center -- an American development company, certified by both State and Federal agencies (in California at least) who promises to build things with your money. They take that money, spend it on development. It's crap shoot as to whether you'll get even a return on that investment because many developers/managers see that money as being "free". Their clients don't care if they get a return, they care about a US citizenship.

What a lot of people don't realize is that the EB5 program doesn't give those foreign investors "real citizenship". EB1 through EB4 are different programs which allow immigration via "Extraordinary ability". Whether you're a famous professor coming from Germany, a highly valued programmer from Indonesia, or a well-known TV personality, those programs allow you to apply for immigration. EB5, similarly, allows investors who have already invested 0.5-1 mil USD to seek the right to immigrate into the United States. What that translates to is this: if you are successful in your bid (which, like most TV stars or famous professors, you will be) you will get a 2-year green card which gives you the right to hold residence in the United States.

What that green card also gives you is the ability to sign all of your assets over to the US government. When you take that green card, the US states that they have the right to tax you and any of your assets...worldwide. Those taxes include income taxes, capital gains taxes, and estate taxes. For the average person who can afford to pay half a million dollars for this sort of thing, they likely fall into a very high tax bracket. On average, that's around 40-50% (federal+state) of their total income/gains/assets paid to the US government. On top of that, if the government decides to not renew your green card (due to the business failing or for other reasons), you will still be taxed for at least 10 more years. Of course, you could refuse to pay, but that just means you'll never be able to do business with an American-owned bank or set foot in American territory again.

Like I mentioned at the beginning, the limitations are well known and that has deterred many people from seeking entry into the US via the EB5 program. The reason why it's becoming fairly popular now is twofold:

1. Regional centers are finally becoming more and more prevalent (they take 3-5 years to set up properly because of regulations and permits and reviews)

2. Many of the EB5 participants are immigrating from China where American education (in general) is perceived to be of high value. You are guaranteed entry into a very good public high school (Gunn, Paly, etc) as long as you live in that district. Not only that, but the UC system is far easier to get into as a "resident" of California.

Anyways, I really wouldn't call it "selling citizenship". It's not a full citizenship. It comes with an asston of strings attached.

Source: I helped set up a regional center in California.