Cohen's lawyer, Lanny Davis, said at the time that "Donald Trump directed [Cohen] to commit a crime by making payments to two women for the principal purpose of influencing an election."
Davis declined CNBC's request for comment.
The Journal reported that Trump, as a presidential candidate, "directed deals in phone calls and meetings" related to the women with Cohen. The U.S. attorney's office in Manhattan, which prosecuted Cohen, has reportedly also collected evidence of Trump's involvement in the agreements.
Daniels' lawyer, Michael Avenatti, told CNBC that the Journal's report "confirms what we have been saying all along."
In a subsequent tweet, Avenatti wrote: "In light of the sworn testimony and evidence relating to the payment to my client and Trump's involvement (confirming our allegations), we are calling for the immediate indictment of the president. No one is above the law in the United States."
Neither the White House, nor Trump's lawyer Jay Sekulow, immediately returned CNBC's requests for comment on the report.
The Journal's story begins with an anecdote about Trump meeting with his friend, American Media Chief Executive David Pecker, in Trump Tower in New York City in 2015.
The report said Trump asked Pecker what he could do to help Trump's presidential campaign.
Pecker then offered to have his supermarket tabloid newspaper, The National Enquirer, "buy the silence of women if they tried to publicize sexual encounters with Mr. Trump," The Journal reported.
Pecker's company later paid a Playboy model, Karen McDougal, $150,000 in return for her story. The Journal said that payment was made after "Mr. Trump asked Mr. Pecker to quash" McDougal's story.
McDougal was freed from her own nondisclosure contract with American Media in April, following a lawsuit in which she claimed that Pecker's company misled her about the terms of the agreement.
Trump's role in arranging the payoffs to Daniels and McDougal were detailed in an indictment that federal prosecutors in Manhattan were preparing to file against Cohen at the time he pleaded guilty.
Since that plea Cohen has met with prosecutors in Manhattan as well as with investigators from the office of special counsel Robert Mueller, who is investigating the Trump campaign's possible collusion with Russians.
Watch: Mueller's investigation could make money for taxpayers