The newspaper, citing a White House official, said Kelly is convinced that Pruitt, who has recently been the subject of reports scrutinizing his spending habits, must step down.
Trump, however, is happy with the job Pruitt has done as the head of the EPA and does not want him to resign, the Journal reported.
In a tweet Friday morning, Trump said that Pruitt is "doing a great job but is TOTALLY under siege."
He reiterated his confidence in Pruitt in remarks to reporters on Thursday.
At a press briefing the next day, press secretary Sarah Huckabee Sanders said that the White House is "continuing to review any of the concerns we have, and we'll keep you posted if there's anything further on that front."
Sanders added: "He has done a good job of carrying out the goals of the EPA."
The Journal's report comes amid a series of controversies surrounding Pruitt's conduct as a public official. Among them are questions about a apartment Pruitt rented in Washington from the wife of an energy lobbyist last year, for which he paid $50 a night.
An EPA watchdog initially cleared Pruitt of any wrongdoing, but later said it was missing certain facts in its initial determination.
The New York Times also reported on Thursday that multiple EPA officials were demoted or reassigned after questioning Pruitt's spending conduct.