Police cars and prison trucks entered and left the Lefortovo prison on Thursday but it was unclear if they carried any passengers and no officials were available to comment.
A number of television cameras and photographers jostled for the best position outside the prison in anticipation of what would the the largest spy swap since the Cold War.
Dmitry said his brother could be sent to Vienna, then onto London, as early as Thursday.
Russia's Interfax news agency cited a rights activist as saying that Sutyagin had already arrived in Vienna on Thursday.
But the activist, Ernst Chorny, told the Reuters news agency that he was not certain it was true.
He said Sutyagin's father had received a call from a Western television channel and was told Sutyagin had been seen arriving in Vienna and being met by a British officer.
Dmitry said his brother only remembered one other person on the list of people to be exchanged for the US suspects - Sergei Skripal, a colonel in Russian military intelligence who in 2006 was sentenced to 13 years on charges of spying for Britain.
In New York, a federal court was to decide the fate of those 10 suspects later Thursday at one hearing.
Five of the suspects had been hurriedly ordered to New York on Wednesday, joining five others already behind bars there.
"I feel our discussions will probably be resolved by tomorrow one way or another," Robert Baum, a lawyer for Anna Chapman, one of the suspects, told The New York Times late on Wednesday.
Nick Spicer, Al Jazeera's correspondent in Washington, said: "We are looking for some kind of information as to if a plea bargain is entered into by the lawyers of the suspects who have told the media they believe some kind of swap is in the offing.
"There aren't any spies known to be in [Russian] jails from the United States, so it's not really quite clear who'd be exchanged.
"Whether this one person [Sutyagin] would be exchanged for the ten, if there might be other researchers, there at least four who have been reported in the media as being possibly involved.
"We just really don't know as the officials in both countries are staying mum."
But the machinations, including a meeting in Washington between US officials and the Russian ambassador on Wednesday, had all the hallmarks as the two former Cold War antagonists moved to scale down tensions stirred up by the US arrests last month.
|A lawyer for Chapman said he expected the situation would be resolved on Thursday [AP]
"A swap seems very much on the cards. There is political will on both sides, and actually by even moving it as far as they have, Moscow has de facto acknowledged that these guys were spies," Pavel Felgenhauer, an intelligence analyst, said.
The alleged spies captured in the US are accused of being "deep cover" agents for Russia's SVR intelligence agency who were tasked with infiltrating US policymaking circles.
They were not assigned to collect classified, secret information, a justice department official said.
They were instead apparently tasked to learn about a broad swath of topics including nuclear weapons, US arms control positions, positions on Iran, White House rumours, CIA leadership turnover, the last presidential election, congress and political parties.
This prompted the authorities to charge them with acting as illegal agents of a foreign government, rather than espionage.
Under the present charges the suspects could face up to 25 years in prison.