Updated

The U.S. may soon deport 10 alleged Russian spies and is attempting to "get rid of" their case as quickly as possible, a lawyer for one of the defendants told Fox News, as a prisoner swap agreement appeared to emerge that could free a nuclear researcher incarcerated in Russia on charges of spying for the U.S.

Members of an alleged network of spies for Russia, which was rounded up on June 27 after years of FBI investigations, could plead guilty to lesser charges and face little jail time -- or be sent directly home to Russia to avoid a diplomatic dustup, Fox News confirmed.

Genesis Peduto, who represents defendant Juan Lázaro, says she expects to get "papers on the deal" as soon as today. She told Fox News she believes the U.S. government wants to "get rid of this case as soon as possible."

Five suspects in the spy case were hastily ordered to New York -- where the five remaining suspects are already waiting in custody.

A scheduled court hearing in Alexandria, Va., for Michael Zottoli, Patricia Mills and Mikhail Semenko was canceled and the trio was ordered to New York. In Boston, defendants Donald Heathfield and his wife, Tracey Lee Ann Foley, waived their right to identity and detention hearings there and were being sent to New York as well.

Earlier, the brother of a man serving a 14-year prison sentence in Russia for alleged spying told reporters in Moscow that the United States and Russia are working on a spy swap.

In Washington, the third-ranking U.S. diplomat, Undersecretary of State for Political Affairs William Burns, a former American ambassador to Moscow, had a Wednesday morning meeting scheduled with the Russian ambassador, Sergey Kislyak. Officials would not discuss the reason for the meeting, the location of which was identified only as "Washington, D.C."

Dmitry Sutyagin said his imprisoned brother Igor was told by Russian officials that he would be released and sent to Britain in exchange for an unknown number of spies. The officials met Igor Sutyagin on Monday at a prison in Arkhangelsk, in northwestern Russia, and U.S. officials were at the meeting, his brother said.

Igor Sutyagin was arrested in 1999 and convicted in 2004 on charges of passing information on nuclear submarines and missile-warning systems to a British company that investigators claimed was a CIA cover.

He said he was made to sign a confession Wednesday morning, although he maintains his innocence and does not want to leave Russia, his brother said. After the meeting, Sutyagin was transferred to Moscow's Lefortovo prison, his brother said.

The Russian Foreign Ministry and the Federal Penitentiary Service said they had no comment on the claim and a spokesman for the U.S. Embassy was not immediately available for comment.

Sutyagin denied that he was spying, saying the information he provided was available from open sources. His case was one of several incidents of Russian academics and scientists being targeted by the Federal Security Service and accused of misusing classified information, revealing state secrets or, in some cases, espionage.

The United States last week arrested 10 people in an alleged spy ring that prosecutors say for the last decade has engaged in secret global travel with false passports, secret code words, fake names, invisible ink and encrypted radio. The spies were allegedly trying to obtain information about American business, scientific and political affairs. They have been charged with acting as unregistered foreign agents.

The arrests came just days after Russian President Dmitry Medvedev visited President Obama during a freewheeling and friendly trip in Washington. Though both the U.S. and Russia said the case would not damage bilateral relations, the arrests clearly nettled the Russian foreign ministry. Plea deals and deportation would speed up the case avoid a public legal battle that could fray nerves and relations in Washington and Moscow.

An 11th suspect was detained in Cyprus last week, but disappeared after being released on bail, triggering a wide manhunt by embarrassed Cypriot authorities.

The U.S. government has opposed the release on bail of any of the defendants, saying they would flee if they had the opportunity.

Fox News' David Lee Miller and the Associated Press contributed to this report.