3 May 2014
Last updated at 18:21 ET
The Sinn Fein leader, who denies any involvement in the killing, is spending his fourth night in police custody.
Police have until 20:00 BST on Sunday to charge or release Mr Adams. The source said there would need to be a significant development for a charge.
If charged he would appear at a special court sitting on Sunday or Monday.
Mrs McConville, a mother of 10, was murdered in 1972.
On Friday, a judge gave police an extra 48 hours to question Mr Adams.
Northern Ireland's Deputy First Minister, Sinn Fein's Martin McGuinness, has said Mr Adams's arrest was part of an effort by some police officers to "settle old scores, whatever the political cost".
Speaking at a rally in Belfast on Saturday, he referred to "an embittered rump of the old RUC (Royal Ulster Constabulary)".
The RUC was replaced in 2001 by the Police Service of Northern Ireland (PSNI).
Mr McGuinness said: "Allegations contained in books and newspaper articles which the PSNI are presenting to Gerry as evidence that he was in the IRA in the 1970s have been around for 40 years.
"But they are only now trying to use these. Is this not political policing?
"This is a replay of the failed effort in 1978 to charge Gerry with membership [of the IRA]."
On Friday, Mr McGuinness hinted that Sinn Fein may look again at whether it would continue to support the PSNI.
The party has claimed the arrest was deliberately timed ahead of elections in three weeks' time.
But the Democratic Unionist Party (DUP) said it showed no-one was above the law.
Secretly buried
Gerry Adams 'questioned for 17 hours a day' over McConville murder
Gerry
Adams is being questioned for up to 17 hours a day by detectives
investigating the murder of Jean McConville, a source close to him says.
Police have until 20:00 BST on Sunday to charge or release Mr Adams. The source said there would need to be a significant development for a charge.
If charged he would appear at a special court sitting on Sunday or Monday.
Mrs McConville, a mother of 10, was murdered in 1972.
On Friday, a judge gave police an extra 48 hours to question Mr Adams.
Northern Ireland's Deputy First Minister, Sinn Fein's Martin McGuinness, has said Mr Adams's arrest was part of an effort by some police officers to "settle old scores, whatever the political cost".
Speaking at a rally in Belfast on Saturday, he referred to "an embittered rump of the old RUC (Royal Ulster Constabulary)".
The RUC was replaced in 2001 by the Police Service of Northern Ireland (PSNI).
Mr McGuinness said: "Allegations contained in books and newspaper articles which the PSNI are presenting to Gerry as evidence that he was in the IRA in the 1970s have been around for 40 years.
"But they are only now trying to use these. Is this not political policing?
"This is a replay of the failed effort in 1978 to charge Gerry with membership [of the IRA]."
On Friday, Mr McGuinness hinted that Sinn Fein may look again at whether it would continue to support the PSNI.
The party has claimed the arrest was deliberately timed ahead of elections in three weeks' time.
But the Democratic Unionist Party (DUP) said it showed no-one was above the law.
Secretly buried
Mrs McConville, a 37-year-old widow, was abducted and shot by the IRA.
Her body was recovered from a beach in County Louth in 2003.
She is one of Northern Ireland's Disappeared, those who were abducted, murdered and buried in secret by republicans during the Troubles.
She was kidnapped in front of her children after being wrongly accused of being an informer to the British Army.
Last month, Ivor Bell, 77, a leader in the Provisional IRA in the 1970s, was charged with aiding and abetting the murder, and there have also been a number of other arrests recently.
The case against Mr Bell is based on an interview he allegedly gave to researchers at Boston College in the US.
The Boston College tapes are a series of interviews with former loyalist and republican paramilitaries, designed to be an oral history of the Troubles.
The paramilitaries were told the tapes would only be made public after their deaths.
But after a series of court cases in the US, some of the content has been handed over to the authorities.
At least one interviewee implicated Mr Adams in the murder of Mrs McConville.
Mr Adams is the former MP for West Belfast and is currently an elected representative for County Louth in the Republic of Ireland.
Her body was recovered from a beach in County Louth in 2003.
She is one of Northern Ireland's Disappeared, those who were abducted, murdered and buried in secret by republicans during the Troubles.
She was kidnapped in front of her children after being wrongly accused of being an informer to the British Army.
Last month, Ivor Bell, 77, a leader in the Provisional IRA in the 1970s, was charged with aiding and abetting the murder, and there have also been a number of other arrests recently.
The case against Mr Bell is based on an interview he allegedly gave to researchers at Boston College in the US.
The Boston College tapes are a series of interviews with former loyalist and republican paramilitaries, designed to be an oral history of the Troubles.
The paramilitaries were told the tapes would only be made public after their deaths.
But after a series of court cases in the US, some of the content has been handed over to the authorities.
At least one interviewee implicated Mr Adams in the murder of Mrs McConville.
Mr Adams is the former MP for West Belfast and is currently an elected representative for County Louth in the Republic of Ireland.
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