The DUP has said it will oppose anyone with a paramilitary conviction becoming the minister responsible for policing or justice if Stormont is restored.
The party has also lobbied for retention of the police full-time reserve in defiance of the Patten Commission recommendations.
Speaking after a meeting with the Northern Secretary, Mr Paul Murphy, at Hillsborough yesterday, the DUP deputy leader, Mr Peter Robinson, warned that "a convicted terrorist" would not be acceptable as a minister for justice.
This means that North Belfast Assembly member Mr Gerry Kelly, or any other elected representative with a prison record, would be opposed by the largest unionist party in any reconstituted Executive at Stormont.
Mr Robinson, who headed a delegation including Mr Nigel Dodds and Mr Jeffrey Donaldson, said: "It would not be acceptable to have someone who is a convicted terrorist as a minister for policing and justice. It is as simple as that."
He added: "I think the essential element for policing and justice powers to be devolved is that the community has confidence in the way they are being exercised and in the people who will exercise that power. That is critical."
Mr Dodds added: "It would certainly not create confidence if we were to have someone like Gerry Kelly or anyone else who has had a terrorist conviction."
Sinn Féin responded quickly, claiming it was not up to the DUP to seek to bar anyone from any ministerial position.
Mr Conor Murphy, leader of the Sinn Féin Assembly group, said: "In our discussions over the summer with the two governments, we have focused on the key issues which we believe all of the participants have a contribution to resolve. This includes the transfer of powers on policing and justice. The DUP have now accepted the logic of the need to transfer policing and justice powers. But it is not down to the DUP to dictate to Sinn Féin or any other parties who to nominate to any future Executive."
The DUP also lobbied yesterday for retention of the full-time reservists, about 1,600 officers who support the PSNI. Unionists fear the police will be hampered by staff shortages if the reservists do not have their contracts renewed next year.
Nationalists want them to be phased out in strict accordance with Mr Patten's recommendations.
Mr Murphy also met the Ulster Unionists yesterday, as a renewed effort to break the political logjam in the North got under way. Fresh bilateral talks with the Northern parties and involving the British and Irish government start at Stormont next week.
This will be followed by a meeting between the Taoiseach, Mr Ahern, and Mr Tony Blair.
The talks will culminate in three days of intensive peace progress negotiations at Leeds Castle in Maidstone, Kent, on September 16th to 18th, involving the two premiers and all the parties.
The DUP has made it clear that next month's talks will need to achieve consensus on a great deal more than the issues of ongoing paramilitarism, illegal weapons and the continuation of the IRA and other groups.
The party also wants agreement on new Assembly structures to allow for greater accountability of ministers to the Assembly.
Speaking of the scale of agreement that is being sought, Mr Dodds said: "We are approaching the coming weeks with a lot of vigour and energy to try and get progress. If the deal is not the right deal, it will not be done. If it is the right deal, it will be done."
The Northern Secretary spoke on Monday of his confidence that both the DUP and Sinn Féin are approaching the next few weeks of critical negotiations in a genuine search for consensus.
"I think there is a seriousness amongst all parties coming to the table that they want to address the issues and they want devolution restored," he said.
However, the governments wish to see consensus on a broad swathe of contentious issues and this may take longer to achieve than by mid-September.