Mr Hume - who with Gerry Adams, the Sinn Fein president, forged an initiative which was a catalyst for the Downing Street Declaration - said he was dismayed that no clear signal had emerged that republicans intended to call a halt to the violence, adding that the party's refusal to endorse the declaration left him disappointed.
But speaking to BBC Radio Ulster, in an interview from his Bordeaux holiday cottage, he said that he hoped a debate was going on within the organisation and that it would very shortly call a ceasefire.
As his colleagues stepped up pressure on him to reveal details of his secret talks with Mr Adams, Mr Hume posed a number of questions for the Sinn Fein and IRA leaderships in the light of recent statements they have made.
'The IRA last week . . . said they were committed to the peace process and their objective was the right of the Irish people to self-determination,' he said.
'My question to them is: 'do you believe that the people of Ireland have that right, and that it is the people of Ireland who are divided about that right, and therefore a solution can only come about through agreement?' '
Mr Hume went on to say that he believed a framework laid out by the Downing Street Declaration was already in place to help achieve that particular agreement and that the people of Ireland were anxious to reach a settlement.
'It is clear that the vast majority of people of Ireland do not want any form of violence. My call to them, which is a direct call, is to respect that and to let us know that they are going to lay down their arms,' he said.
'I have given total commitment to the entire peace process, and I believe the time has come for the IRA to show that they are genuinely committed to peace and to understand that a peace process is also a peaceful process.'Reuse content