Constable Carroll, 48, was shot dead after he responded to a 999 emergency call in Craigavon in March 2009.
At the time the dissident republican group, the Continuity IRA, claimed it was responsible for the shooting.
Brendan McConville and John Paul Wootton were found guilty of murder last year, but are both appealing against their convictions.
McConville is from Glenholme Avenue in Craigavon, while Wootton, who was 17 at the time of the murder, is from Collindale in Lurgan, also in County Armagh.
A barrister acting for Brendan McConville told the Court of Appeal that there were “discrepancies” in the account of a witness who had told the original trial that he had seen his client near the scene of the shooting on the night of the murder.
The man, known as Witness M, had given his account to police 11 months after the fatal gun attack on Constable Carroll.
The barrister told the court that Witness M's account had not been given in a measured, reliable or consistent fashion.
He also said the Court of Appeal would hear from a new witness, who will say that Witness M “lives in world of his own” and often said things that simply were not true.
Constable Carroll was the first officer to be killed in Northern Ireland since the formation of the Police Service of Northern Ireland in 2001.
