Not sure exactly what you mean here by "the media", but the US has financed (and continues to finance) some media, such as VOA or RFERL, typically for foreign audiences. Under the present organization both of these (and a few other more obscure outlets) belong to the U.S. Agency for Global Media (USAGM), which had an annual budget of some $753 million (FY 2016) and some 3,500 employees, so not exactly small potatoes. For comparison, CNN has some 4,000 employees, although probably more revenue/budget (one figure I saw was $1.2B in revenue for 2018.)
A 1948 law prohibited such stations like VOA and RFERL from broadcasting mainly to Americans. Also, throughout much of the Cold War, these belonged to the better known and better funded (around $2B/year) USIA.
Some countries like Russia responded in kind to the US requirement that RT register as a foreign agent, requiring (in 2017) VOA [& RFERL] to do the same etc.
Moscow's action [...] came in retaliation to Washington's move [...] forcing a U.S.-based affiliate of Kremlin-funded RT, formerly known as Russia Today, to register under the Foreign Agents Registration Act, also known as "FARA." The act requires people acting as agents of foreign governments to publicly disclose their relationship. [...]
Russian officials have called the new legislation a "symmetrical response" to what they describe as U.S. pressure on Russian media.
As far as the PR budget of the US gov't, I don't exactly what that was about as the link from Fox News' reporting is a dead link now. The US government isn't limited to financing those USAGM outlets though, as far as PR efforts go. The Fox News story mentioned that "The GAO reported that the spending is concentrated among a handful of agencies, especially the Defense Department." So, that's almost certainly another bucket of money. From the same story:
The report noted that agencies “may have legitimate interests in communicating with the public” and gave several examples – such as the IRS providing information about a tax credit or the Education Department providing information about student aid applications.
So that's a different issue than taking adverts in the [domestic] media or sponsoring media outlets for foreign policy purposes.
According to a Reason article from 2016 that's almost certainly talking about the same report:
that includes everything from press releases and safety bulletins to television ads for Obamacare, direct mailers about the importance of getting flu shots and endless streams of tweets and Facebook posts intended to connect the average American with their government.
Every department and agency in the government does it, but no one does it as much as the Department of Defense.
The Pentagon accounted for 60 percent of all public relations spending between 2006 and 2015, the GAO found, and it employs about 40 percent of the more than 5,000 public relations workers in the federal government.
For context: there are only 4,500 employees in the U.S. Department of Education.
So that report did not seem to include USAGM under "PR". Perhaps the [accounting] subtlety is (given figure 6 quoted from the report) that that GAO
report is only about "advertising and public relations contracts"--emphasis mine; since USAGM
is funded by a line item in the budget and not by a contract... it doesn't count under (any) "contracts".