Whelan was arrested by Russia's Federal Security Service on Dec. 28. His family have said he is innocent and that he was in Moscow to attend a wedding.
The ministry said the United States detained Russian citizen Dmitry Makarenko on the Northern Mariana Islands on Dec. 29 and had moved him to Florida.
"... Makarenko, born in 1979, has arrived on Saipan Island with his wife, underage children and elderly parents. He was detained by FBI personnel at the airport right after his arrival," the ministry said.
The US Embassy in Moscow could not immediately be reached for comment.
The accusations from both sides could further complicate a strained relationship between Moscow and Washington, despite the professed desire of Presidents Donald Trump and Vladimir Putin to build a personal rapport.
US Secretary of State Mike Pompeo said this week Washington had asked Moscow to explain Whelan's arrest and would demand his immediate return if it determined his detention is inappropriate.
Britain cautioned Russia on Friday that individuals should not be used as diplomatic pawns. Whelan also holds a British passport.
The Russian ministry said in its statement Moscow diplomats had not been able to reach Makarenko in Florida and said Washington had yet to explain his detention.
Before Moscow gave details of Makarenko's detention, experts had speculated that Moscow could exchange Whelan for Russian nationals held by Washington.
Commenting on that possibility, Russian Deputy Foreign Minister Sergei Ryabkov said on Saturday:
"I see no reasons to raise this issue in context of exchanges. We should undergo all the procedures needed in this situation," Interfax news agency quoted Ryabkov as saying.