Michael Avenatti, Daniels' lawyer, also said that Trump and Cohen should immediately release bank statements that could shed light on a series of payments made to Cohen's shell company.
"This is an enormous amount of money," Avenatti said on MSNBC's "Morning Joe" show.
"You've got millions of dollars that are ... being deposited into this account," Avenatti said. "Michael Cohen appears to be selling access to the president of the United States."
Later Wednesday, a senior official at Swiss pharmaceuticals giant Novartis told NBC News that Cohen had reached out to Novartis shortly after Trump's election "promising access" to the new administration. Cohen and his attorney did not respond to a request for comment to NBC News.
White House press secretary Sarah Huckabee Sanders, when asked Wednesday if Trump was worried about major corporations paying his lawyer, said, "I haven't heard the president expressed any specific concerns about that."
An estimated $4.4 million or more has been identified as flowing through Cohen's shell company, Essential Consultants, which was created in October 2016, a month before Trump's election, according to a report by Avenatti's firm.
Cohen said Wednesday that Avenatti's "document is inaccurate," according to NBC News. He did not elaborate on what was supposedly wrong about the document.
Also on Wednesday, Avenatti scoffed at the claims being made by the various companies about why they paid Cohen.
Avenatti on Tuesday released an investigatory report that says Essential Consultants received about $500,000 from a company that Avenatti claims is controlled by a Russian oligarch, as well as payments from pharmaceuticals giant Novartis, telecommunications behemoth AT&T and defense contractor Korea Aerospace Industries.
In an AT&T email to employees, which was obtained by CNBC, the company said it hired Cohen as one of several consultants in the wake of Trump's election. Here is the memo in full:
"From: T Now
Sent: Wednesday, May 09, 2018 12:10 PM
Subject: Perspective on the news
Perspective on the news
To: All U.S. AT&T employees
Late yesterday, many media outlets reported that in 2017, AT&T hired Michael Cohen, a former lawyer with the Trump Organization. We want you to know the facts.
In early 2017, as President Trump was taking office, we hired several consultants to help us understand how the President and his administration might approach a wide range of policy issues important to the company, including regulatory reform at the FCC, corporate tax reform and antitrust enforcement. Companies often hire consultants for these purposes, especially at the beginning of a new Presidential Administration, and we have done so in previous Administrations, as well.
Cohen was one of those consultants. Cohen did no legal or lobbying work for us, and our contract with Cohen expired at the end of its term in December 2017. It was not until the following month in January 2018 that the media first reported, and AT&T first became aware of, the current controversy surrounding Cohen."
"We now have multiple different things supposedly that Michael Cohen was doing for all these companies," Avenatti said. "Now we hear from Novartis that he was hired on health-care matters. Evidently, he's a doctor. One of the companies mentioned they hired him for real estate matters. He's a real estate agent. Another company stated that they hired him for accounting advice. Evidently, he's an accountant."
Avenatti's comments came hours before Novartis revealed it had paid Cohen up to $1.2 million for work that he quickly proved unable to do. That is three times what Avenatti's probe had found Novartis gave Cohen.
Within weeks of being created by Cohen in October 2016, Essential Consultants was used to pay Daniels $130,000 in exchange for her agreement to not publicly discuss an alleged sexual encounter she claims she had with Trump, the actress has said.
Cohen confirmed he paid the money, but the White House denies any such affair. Rudy Giuliani, an outside attorney representing Trump in the special counsel's Russia probe, has said that Trump reimbursed Cohen for his porn star payoff.
Avenatti's new report detailed payments to Essential Consultants that were not previously known.
On Wednesday, Reuters reported, quoting a source, that AT&T paid Cohen more than the $200,000 identified by Avenatti's report. The company confirmed to CNBC again on Wednesday that it made the payments in exchange for "insights" into the Trump administration, but did not detail the total amount paid. Dow Jones, citing a source, reported that AT&T paid Cohen's company up to $600,000.
Columbus Nova, the investment management firm that Avenatti alleges is controlled by Russia oligarch Viktor Vekselberg, disputed his claim, saying the firm is "solely owned and operated by Americans."
The firm also has claimed that it hired Cohen as a consultant for possible sources of capital and potential real estate investments and other ventures.
The "claim that Viktor Vekselberg was involved in or provided any funding for Columbus Nova's engagement of Michael Cohen is patently untrue," an attorney for Columbus Nova said.
Novartis says that it was using Essential Consultants for consulting on issues related to American health policy. The company also said it was contacted last November by special counsel Robert Mueller's office, and "provided all the information requested."
Mueller is investigating Russian interference in the 2016 presidential election, and possible collusion by members of Trump's campaign in that meddling.
Novartis in a statement to CNBC said it had paid $100,000 per month to Essential Consultants under the terms of the contract signed in February 2017, despite the fact that the company quickly learned that Cohen was unable to do the work expected.
Novartis said it had to continue paying him until the contract ended this past February because the agreement could only be terminated for cause. While the company did not reveal how much Cohen ended up being paid in total, it could have been $1.2 million.
A KAI spokesman said it signed a contract with Essential for "legal consulting concerning accounting standards on production costs."