The release of the memo raised fresh questions about whether Mr. Trump — who last year fired James B. Comey, the F.B.I. director overseeing the Russia inquiry, and then sought to remove Robert S. Mueller III, the special counsel named to take it over — might seek next to oust Rod J. Rosenstein, the deputy attorney general, who named Mr. Mueller.

The president, who first considered getting rid of Mr. Rosenstein last summer, pointedly refused to say on Friday whether he was more likely to do so now, cocking his head and telling reporters who pressed him on the matter: “You figure that out.”

But the release of the memo underscored how Mr. Trump has transformed his own suspicions and unsubstantiated theories about an inquiry he has repeatedly called a “hoax” and a “witch hunt” into a set of official accusations of corruption against the very people investigating him. The president has called for months for the compilation of such evidence, often taking to Twitter to demand that the Justice Department and the F.B.I. release information that could show political bias on the part of those investigating whether his campaign colluded with Russia.

Last month, Representative Devin Nunes, the California Republican who leads the Intelligence Committee and is a loyal ally of Mr. Trump, obliged by dispatching his staff to compile a classified document that seeks to illustrate just that. Mr. Trump was eager to release it.

The memo described Mr. Rosenstein as one of the senior Justice Department officials who approved an application to extend surveillance of Mr. Page, and suggested that those applications deliberately avoided mentioning that they were based in part on information in a dossier paid for by Democrats.