The Department of Justice's Deputy Attorney General Rod Rosenstein appointed Robert Mueller, an American lawyer and former FBI director, as special counsel to the Russian Federation investigation in May.
Trump earlier derided claims of collusion between Moscow and his associates as "phony" Thursday after reports the head of a probe into Russia's alleged election meddling is investigating him for obstructing justice.
A day earlier, Trump had questioned why his general election rival Hillary Clinton was not being investigated.
"I think he's looking for information that would bolster the case that Comey made publicly and in his memos - that he believed Trump had directed him to drop the investigation into Michael Flynn, who was President Trump's first national security adviser".
Trump later told NBC's Lester Holt he had planned to fire Comey regardless of what recommendation he received from Rosenstein and Attorney General Jeff Sessions, who has recused himself from the Russian Federation investigation.More news: Rosenstein may need to recuse himself from Russian Federation probe
Kasowitz said on Wednesday that leaking Federal Bureau of Investigation information about the president was "outrageous, inexcusable and illegal", although he did not question the content of the article.
As recently as Monday morning, the panel on MSNBC's "Morning Joe" ripped Trump for his continuous rants on Twitter, with co-host Mika Brzezinski saying, "He can't help himself".
While accusing the Clinton campaign and the Obama administration of indulging in illegal acts, Trump had said that "there was never a special counsel appointed to oversee the acts of his Democrat predecessors". "I hope you can let this go".
Seth Moulton, a congressman from Salem in Massachusetts, tweeted, "As the Representative of Salem, MA, I can confirm that this is false", referring to Mr Trump's "witch hunt" whine.
Mr Trump is unlikely to face criminal prosecution as a sitting president, but obstruction of justice could be grounds for impeachment.
While Trump did not identify Rosenstein by name, the former US attorney, whom Trump nominated, is overseeing the ongoing Russian Federation probe.