Moscow and Washington exchanged a flurry of rebukes over the detention of a United States citizen who Russian authorities said was suspected of espionage.

Russia has allowed consular access to American Paul Whelan, who was detained in Moscow on Dec 28, the Russian Foreign Ministry said on Wednesday.

Whelan was taken into custody during an "act of espionage in Moscow", the Federal Security Service said on Monday.

A criminal investigation was opened against him, and if found guilty of spying, he might be sentenced to 10 to 20 years in prison.

Whelan is the head of global security for a Michigan-based auto parts supplier, according to the Associated Press. The 48-year-old was in Moscow to attend a wedding when he disappeared, his brother, David Whelan, said on Tuesday.

The US State Department said that it was notified of the detention of Whelan, requested consular access and expected "Russian authorities to provide it".

Russian Foreign Ministry spokeswoman Maria Zakharova said on Wednesday that the government provided consular access to Whelan, and he was visited by US Ambassador Jon Huntsman, who expressed his support and offered the embassy's assistance, the State Department said in a statement later that day.

Secretary of State Mike Pompeo said, "If the detention is not appropriate, we will demand his immediate return."

Pompeo said the Trump administration has "made clear to the Russians our expectation that we will learn more about the charges and come to understand what it is he's been accused of".

Whelan's family posted a statement on Twitter, "We are deeply concerned for his safety and wellbeing. His innocence is undoubted, and we trust that his rights will be respected."

David Whelan said in an interview that his brother had been to Russia several times, so when a fellow former Marine was planning a wedding in Moscow, he was asked to go along to help out.

He said he learned of his brother's arrest while searching the internet on Monday.

The detention came after Russian citizen Maria Butina pleaded guilty in a Washington court on Dec 13 to one conspiracy charge, admitting that she acted as an unregistered foreign agent to advance Moscow's interests, according to Xinhua News Agency.

Butina, 29, was arrested on July 15 and charged with conspiracy to act as an agent of the Russian government. In Moscow, the Foreign Ministry described charges against Butina as "false accusations".

The arrest was announced on July 16, only hours after US President Donald Trump finished a summit meeting and a news conference with his counterpart, Vladimir Putin, in Helsinki, Finland.

Putin said that Butina has no ties to Russian intelligence services, and that the whole case against her was unfounded, according to Russia's TASS News Agency.