Lawyers for two men attempting to overturn their convictions for the killing are seeking an order for the Criminal Cases Review Commission to examine the arrest of a new witness in the case.
They are also making a separate application to the Police Ombudsman to look into allegations of misconduct in the operation.
Last week judges were told police have tried to sabotage appeals being mounted by John Paul Wooton and Brendan McConville.
Defence counsel made the claim as their joint challenge to being found guilty of the murder was put back until October amid uncertainty over potential fresh evidence.
Constable Carroll was ambushed and shot dead as he responded to a 999 call at Lismore Manor, Craigavon in March 2009.
McConville, 41, of Glenholme Avenue in the town, is serving at least a 25-year sentence for the murder.
Wootton, 22, of Collindale, Lurgan, received a minimum 14-year term.
It was revealed in court that a man related to a key prosecution witness was arrested two weeks ago and held for two days before being released without charge.
This man has made a sworn court statement branding his relative a compulsive liar.
According to defence lawyers, police arrested him in a bid to pressure him into withdrawing his evidence, and warned he would be discredited if he testified.
They now want the Court of Appeal to direct the Criminal Cases Review Commission (CCRC) to look into this aspect of the case.
Judghes have listed their application for a hearing next Wednesday.
It was also confirmed on Tuesday that transcripts from all but one of the 11 interviews carried out with this new witness have been handed over to defence teams.
As he updated the court Ciaran Murphy QC, prosecuting, said ongoing police inquiries are expected to take another four weeks.
Lord Chief Justice Sir Declan Morgan pointed out: "Part of this application is about who should conduct (or) supervise that investigation."
Mr Murphy replied by suggesting the CCRC may have difficulties in dealing with a continuing police probe.