He also allegedly lied about his communications with an unknown individual, referred to in the papers as "Person A", and deleted emails between him and "Person A".
Van der Zwaan is expected to appear in court Tuesday afternoon.
Bloomberg puts Mr Khan's net worth at $9.7 billion.
The charge against Zwaan and reports that Gates may plead guilty suggest "that Mueller is putting even more pressure against Manafort, which is interesting because Manafort is already buried in an indictment with very serious charges", Mariotti said. A USA judge set his sentencing for April 3.
Manafort, who was Trump's campaign manager for nearly five months in 2016, and Gates, who was deputy campaign manager, pleaded not guilty previous year to Mueller's charges.
Trump continued to fume about the investigation, blaming former President Barack Obama for failing to take his claims of voter fraud seriously.
Though Manafort was not mentioned by name in the court document about van der Zwaan, the history between Manafort and the law firm suggests that the special counsel has cast a wide net in investigating Manafort's business relationships around the world.
Zwaan, who married the daughter of a Russian-Ukrainian oligarch past year, admitted making false statements, as part of a plea agreement with the special prosecutor on the Russia investigation, Robert Mueller.
Van der Zwaan's plea also illustrates how aggressively the special counsel, Robert Mueller, has pursued those who obstruct his inquiry.
The guilty plea to making false statements to the Special Counsel's Office carries a maximum sentence of 5 years in prison and a $250,000 fine.
The Skadden report found that Yanukovych-era legal officials did not deprive Tymoshenko of her right to due process during the trial, in a report that State Department official Victoria Nuland criticized at the time as "incomplete". He enlisted the firm in his effort to shield a client, Viktor Yanukovych, the Russia-aligned president of Ukraine, from global condemnation.
Skadden issued a $567,000 refund in June 2017 to the Ukrainian government after an investigation revealed that former Justice Minister Oleksandr Lavrynovych embezzled $1.1 million to hire the firm.
Just days later, Mueller's team netted yet another guilty plea, its fourth, this time from Dutch national Alex Van Der Zwaan, 33, who prosecutors say made false statements about communications in 2016 with Gates and an unnamed person. Yanukovych defeated Ukraine's anti-Russian prime minister, Yulia Tymoshenko, in the 2010 presidential election but was ousted during a revolution in February 2014. The man who commissioned the report was Paul Manafort, who served as Trump's campaign manager until he was sacked in August 2016 amid revelations about extensive lobbying work for a pro-Russian Ukrainian political party.
An attorney for worldwide law firm Skadden, Arps, Meagher, & Flom who helped whitewash the prosecution and imprisonment of Batkivshchyna Party leader Yulia Tymoshenko was charged with perjury in USA federal court today.