Mr Condit admitted for the first time that he had an affair with Ms Levy, an intern with the Federal Bureau of Prisons, it was reported on Sunday.
"The family wants the comfort of knowing that the people who were closest to Chandra are giving complete and truthful information to investigators," Michael Frisby, a spokesman for Robert and Susan Levy, told the paper.
Mr Condit admitted for the first time that he had an affair with Ms Levy, an intern with the Federal Bureau of Prisons, it was reported on Sunday.
Ms Levy disappeared on 30 April, but Washington DC police have said that Mr Condit is not a suspect in the investigation.
Mr Condit's lawyer, Washington attorney Abbe Lowell, said that it was not necessary for his client to submit to a lie detector.
When pressed, Mr Lowell said under certain conditions Mr Condit would take a lie detector test.
"If the police said that he was a suspect or that there was some discrepancy, I would like them to come to me," Mr Lowell said.
The police said Mr Condit answered all of their questions satisfactorily after a third interview on Friday night which lasted 90 minutes.
"The congressman was not a suspect before the meeting, he was not a suspect during the meeting and he is not a suspect since the meeting," said Terrance Gainer, Washington's second-ranking police official, at a press conference on Saturday.
Romantic relationship?
Mr Gainer would not, however, reveal whether Mr Condit described his relationship with Ms Levy as romantic.
"With the clarity of this interview and the interview with his wife [last week], I think we have the information we need," he said.
Previously, the 53-year-old California Democrat described Ms Levy as a good friend, and his aides have denied any romantic relationship between them.
But Ms Levy's aunt, Linda Zamsky, told the Washington Post in an interview last week that her niece told her she was dating Mr Condit.
Ms Levy, 24, was last seen when she cancelled her membership of a Washington health club.
It appears she was planning to go to her parents' home, in Modesto, California.
Police went to her apartment in May, after receiving calls from her worried parents. There they found her bags packed, but no signs of violence.
The investigation so far has been handled as a missing persons case, not a crime.