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I saw the President explicitly tell a newspaper that it should reveal its sources:

But for the sake of our national security, the New York Times should publish his name at once. I think their reporters should go and investigate who it is. That would actually be a good scoop.

Is this not illegal in the U.S.?

Where the President explicitly tells a newspaper that they should reveal their sources. Is this not illegal in the US?

It is not illegal. Well, it would be a U.S. Attorney, rather than the President himself.

You're thinking of shield laws, but no such law exists at the federal level. Moreover, although some people might think that the First Amendment ("freedom of the press") would protect a journalist in such a case, the Supreme Court has held that it doesn't, though the government is required to "convincingly show a substantial relation between the information sought and a subject of overriding and compelling state interest". So if a reporter were subpoenaed in federal court to reveal a source, and they couldn't convince a court to quash the subpoena under the Branzburg test, they'd have to either reveal their source or be held in contempt. In practice, many reporters have chosen the latter, and served time in jail rather than reveal the source. For instance, Judith Miller.

There've been a number of proposals to create such a federal law, but so far Congress hasn't seen fit to do it.

According to the Wikipedia article, every state except Wyoming has some sort of shield protection for journalists, either in statute or case law, though the protections are not necessarily absolute. So a state court or prosecutor would find it much more difficult to enforce such an order.

There's no law against him saying the paper should name the source, nor is there any law against directly asking the paper to name its source. The president was at a campaign rally, and his speech is protected by the same First Amendment that protects the New York Times.

At the federal level, the paper could be forced to reveal the source, but it would have to obtain a subpoena and probably spend quite a bit of time litigating over whether that subpoena was legitimate.

Although the law does not require it, Department of Justice guidelines restrict prosecutors' ability to obtain this information. They're fairly detailed, but the most relevant part is here:

Members of the Department must consult with the Criminal Division before taking steps to enforce subpoenas issued to member of the news media, or to compel compliance with subpoenas or court orders issued to third parties for communications records or business records of member of the news media, which subpoenas were issued or court orders obtained in the first instance by other Executive Branch departments or agencies. To satisfy the consultation requirement, members of the Department shall submit to the PSEU a memorandum describing the factual and legal background of the matter. Members of the Department may not proceed with any efforts to enforce or compel compliance with any subpoenas or court orders until the Criminal Division has responded in writing to the request for consultation.

At the state level, there are laws in place protecting reporters from being forced to reveal their sources, but they vary from state to state. New York, Ohio, Alaska, and some others have virtually absolute statutory protection. California, Texas and Florida have somewhat weaker statutory protections. A few states, mostly in New England and the Midwest, have no statutory protection, though courts have established protections based on free-press principles. I believe Wyoming is the only state that follows the federal government's anything-goes approach.