Coulson was found guilty on Tuesday of plotting to hack phones while he was editor of the News of the World.
An Old Bailey jury is still considering allegations that Coulson, 46, conspired with former royal editor Clive Goodman 56, to commit misconduct in a public office by agreeing to pay police officers for two royal directories. They both deny the two charges.
Coulson, who was forced to resign as Cameron's director of communications over the scandal, faces up to two years in jail for hacking following the high-profile trial.
The jury of eight women and three men found him guilty of conspiring to hack phones, but cleared ex-News International chief executive Rebekah Brooks of all charges.
Questions surround phone hacking at News of the World
Married father-of-three Coulson was recruited by Chancellor George Osborne to head up the Tory media operation within months of resigning as News of the World editor in January 2007.
When Cameron entered Downing Street, the former journalist took on duties heading up the Number 10 spin operation, quitting shortly before he was arrested over the phone-hacking scandal.
It is expected that Cameron will be asked about Coulson at Prime Minister's Questions on Wednesday.
Following the verdict on Tuesday, Cameron said: "I take full responsibility for employing Andy Coulson. I did so on the basis of undertakings I was given by him about phone hacking and those turned out not to be the case.
"I always said that if they turned out to be wrong, I would make a full and frank apology and I do that today. I am extremely sorry that I employed him. It was the wrong decision and I am very clear about that."
Osborne said: "We gave him a second chance but, knowing what we now know, it's clear that we made the wrong decision."
Tuesday's partial verdicts were delivered on the jury's eighth day of deliberations and the 138th day of the trial.
Brooks, 46, was cleared of hacking, misconduct in a public office for allegedly signing off payments to a Sun journalist's "number one military contact" between 2004 and 2012, conspiracy to pervert the course of justice and perverting the course of justice.
The jury will resume deliberations from 11am local time on Wednesday.
Rupert Murdoch to be interviewed over phone-hacking scandal
Meanwhile, Media mogul Rupert Murdoch could soon be interviewed by British police over the phone-hacking scandal.
The Guardian newspaper says detectives first contacted Murdoch last year to question him over allegations of crime at his British newspapers.
But they agreed to a request from his lawyers to wait until the long-running phone-hacking trial was finished.
A Metropolitan police spokesman would not discuss the prospect of Murdoch being questioned.
