"They both pleaded not guilty," the US Attorney's Office said on Thursday.
Correia and Kukushkin were charged earlier in the week and are set to make their next appearance in court on December 2 at 2:00 p.m. [6:00 p.m. GMT], the US Attorney's Office added.
Last week, US federal prosecutors unsealed an indictment against Correia and Kukushkin as well as two other businessman - Lev Parnas and Igor Fruman - for funnelling $1 million in illegal donations from foreign nationals to unspecified political campaigns.
Parnas and Fruman made their initial appearance at the US District Court for the Eastern District of Virginia on October 10. Later that day, the two men were released on a $1 million bond each.
In one of several schemes they are accused of carrying out, the four men allegedly conspired to make political donations to US candidates funded by a foreign national, who the indictment identified as "a Russian citizen and businessman who, at all relevant times, was not a citizen or lawful permanent resident of the United States."
In another plot, the four men are accused to have donated money to a member of Congress and then tried to press the lawmaker to "remove or recall the then-US ambassador to Ukraine."
That effort, the indictment alleged, was carried out "at least in part, at the request of one or more Ukrainian government officials."
Bill Sweeney, assistant director in charge of the FBI's New York office, said last week that co-defendant Kukushkin was arrested in San Francisco shortly after Parnas and Fruman were apprehended in Virginia, while Correia remains at large.
Sweeney emphasized that the investigation is ongoing and is focused on uncovering genuine corruption.
The indictment did not mention Giuliani by name, but the lawyer in a May 18 Twitter post identified both Parnas and Fruman as his clients.
Lawmakers in the US House of Representatives have sought to depose both Parnas and Fruman as part of their ongoing impeachment investigation into Trump.