Citing three unnamed people briefed on the subject, the Times said that Mueller, a former FBI director, was scrutinizing tweets and negative statements from the president about Attorney General Jeff Sessions and Mueller's FBI successor James Comey.
Trump has repeatedly turned to Twitter to try to bend proceedings and public opinion in the direction he would like - calling for Mueller's probe to end and demanding that the Justice Department investigate his political enemies.
In one of many similar tweets, Trump on April 20 appeared to try to undermine Mueller's probe into whether the Trump campaign colluded with Russia in its efforts to skew the 2016 US presidential election.
"James Comey illegally leaked classified documents to the press in order to generate a Special Council [sic]? Therefore, the Special Council was established based on an illegal act? Really, does everybody know what that means?" he wrote.
Now Mueller is reportedly investigating whether that pattern of behavior is enough to constitute obstruction of justice. Obstruction is difficult to prove, and US legal opinion is divided on whether the president can be indicted for such a crime even if it is a slam dunk case.