Media reports on Wednesday said Russian police had detained Kinnock, 37, on January 15 for drunk driving and other traffic violations. However, as Kinnock enjoys diplomatic immunity, he will not be charged over the matter.
"Kinnock was handed over to members of the Consulate," the source said.
Stephen Kinnock's father, Neil Kinnock, took up the post of head of the British Council in 2004 after stepping down as vice-president of the European Commission.
The arrest of the son of former British prime minister Margaret Thatcher's one-time bitter political rival came as a row between the British Council, the cultural arm of the British government, and Russia continued.
On Friday, the British Council's office in St. Petersburg froze its projects for the next three weeks amid pressure from Russian authorities over alleged tax arrears, a spokesman said.
The Council's chief executive, Martin Davidson, said on Thursday that 10 employees had received late-night visits from Russian tax police, and that all the 20 staffers had been interviewed by Russian security officials.
British Foreign Secretary David Miliband described Russia's actions as "reprehensible" on Thursday, adding that they were a "stain" on the country's reputation.
Political commentators in Britain are under no doubt that the British Council dispute is inextricably linked to Britain's repeated requests for the extradition of Andrei Lugovoi, the man it suspects of being behind the murder of ex-Russian security services member Alexander Litvinenko in London in November 2006. Russia has refused to extradite Lugovoi, citing its Constitution.
Russia has accused Britain of attempting to politicize the issue, which it says is purely a legal matter.
Konstantin Kosachyov, who heads the international affairs committee at the lower house of Russia's parliament, said no legal impediment would remain to the British Council in Russia as soon as it brought its operations in line with Russian law.
"I guess Russia is the only place where it [the British Council] has been attempting to act in line with British law. Russia is not a banana republic, we have a functioning legal system and any foreign organization, irrespective of the type of activities it carries out, has to comply with Russian laws," the parliamentarian said.