Lawyer: Condit has cooperated fully with police

"We could inadvertently make things harder to find her (Levy) by giving public disclosure of bits and pieces of information that only the police have the right to know," said Condit attorney Abbe Lowell.

WASHINGTON (CNN) -- Californian Rep. Gary Condit has given police "every shred of information to help them try to find" the missing former intern Chandra Levy, Condit's lawyer told CNN on Sunday.

Police sources told CNN that Condit admitted, in his third interview with police, that he had a romantic relationship with Levy. Attorney Abbe Lowell said on CNN's "Late Edition with Wolf Blitzer" that Condit had sought the third interview with police. Lowell also said that Condit has never been under a grand jury threat, as some media reports have said.

Lowell also said there was a very good reason why his client has not spoken with the media about the former intern who has now been missing for two months.

"He has given them (police) every shred of information to help them try to find her," Lowell said. "We could inadvertently make things harder to find her by giving public disclosure of bits and pieces of information that only the police have the right to know."

VIDEO Constituents of U.S. Rep. Gary Condit say they are beginning to question their support for him in the Chandra Levy investigation. CNN's Frank Buckley reports (July 8) Play video (QuickTime, Real or Windows Media)   RESOURCES CNN Access: Condit's attorney Abbe Lowell   Message Board: The case of the missing intern   More     MORE STORIES Condit supporters want him to speak out   Lawyer: Condit feels abused by media   Carolyn Condit questioned about Washington visit

Condit has denied a romantic relationship with Levy through his aides and refers to her as a "good friend."

When questioned whether the congressman would admit to an affair with Levy if one had occurred, Lowell said:

"To whom? To you? To his family? To the police? The congressman has been totally cooperative with the police -- they know all the facts. I hope it helps them find Chandra Levy. Everything else about this is a matter of privacy among the Condit family."

Police: Condit not a suspect

Police sources said Saturday that Condit admitted during the third interview Friday of having a romantic relationship with Levy, something the congressman had denied for eight weeks through his aides.

Levy, a 24-year-old former intern with the Bureau of Prisons, was last reported seen at a Washington gym on April 30, although an e-mail sent from her computer indicates she may have been at her Washington home on May 1. She had been planning to return to her parents' home in Modesto, California.

Executive Assistant Chief Terrance Gainer of the D.C. Metropolitan Police, who would not confirm that Condit admitted an affair, said police are satisfied that Condit answered every question asked of him.

"He was not a suspect before the meeting, he was not a suspect during the meeting and he's not a suspect since the meeting," he said. "So he was cooperative, and happy to get this third interview out of the way."

Gainer said police believe they understand the nature of the relationship between Condit and Levy, but "unfortunately it does not lead us to finding where Miss Chandra Ann Levy is."

He said there are no plans for further interviews with Condit, 53.

'Best chance of finding their daughter'

During his rounds of Sunday talk shows, Lowell blasted the media for prying into Condit's private life by asking about allegations he had an affair with Levy.

When Levy's aunt announced to reporters Friday that her niece had carried on a secret, intimate relationship with Condit since last fall, she spoke of "their bedroom encounters," the gifts her niece said she received and of the lengths they went to keep relationship secret.

But criminal defense lawyer Roy Black told CNN that the Levy family is using a good strategy.

"By focusing on Representative Condit, and then on whether or not he is having an affair, and whether he has had multiple affairs, this of course gets everybody's interest," Black told Blitzer. "It keeps this investigation going, it keeps it on television and this is their best chance of finding their daughter."

Lowell on Sunday disputed news reports that Condit's telephone records had been subpoenaed.

"Now, there never has been a subpoena issued that we know about for anything for congressman Condit," Lowell said.

This week, police plan to take cadaver-sniffing dogs to trash dumps in the Washington area.

-- CNN Correspondent Bob Franken contributed to this report.