Coulson was convicted on Tuesday of conspiracy to hack phones. Goodman had already pleaded guilty in 2006.
But jurors could not decide on charges against the pair of conspiracy to commit misconduct in public office.
The judge criticised David Cameron for commenting on the case while the jury was still deliberating.
Meanwhile, the BBC has learned that one of the phone hacking victims was Carole Middleton, the Duchess of Cambridge's mother.
On Tuesday the prime minister apologised for employing Coulson as his communications chief but that he had done so "on the basis of undertakings I was given by him about phone hacking and those turned out not to be the case."
Coulson's lawyer criticised Mr Cameron's "ill-advised and premature intervention" in the case - a move described as "unwise" by former lord chancellor Ken Clarke.
The judge commented: "I consider that what has happened is unsatisfactory so far as justice and the rule of law are concerned."
Discharging the jury at the Old Bailey, Mr Justice Saunders said a decision on any retrial would be made on Monday.
Five people have pleaded guilty in the case, while former News International chief executive Rebekah Brooks and her husband Charlie were among the remaining defendants cleared on Tuesday.
Mr Cameron faced criticism from Labour leader Ed Miliband at Prime Minister's Questions for his decision to take Coulson with him into Downing Street.
Labour wants an investigation into why Coulson did not receive the highest level of security vetting when he joined the No. 10 operation.
But Mr Cameron insisted it had been the civil service's role to vet the appointment and that he had been exonerated by a public inquiry following the hacking scandal.
"Every single one of these issues was dealt with exhaustively by the Leveson Inquiry," the prime minster told MPs.
"He looked into all of these questions about the warnings I was given and the response I gave and he made no criticism of my conduct."
The BBC's Robert Peston said that, if there was to be an investigation, it would be head of the civil service, Sir Jeremy Heywood, facing embarrassment over the failure to vet Coulson.
However, Mr Miliband listed a string of allegations about the PM's role, insisting: "When it came to Andy Coulson, you just didn't want to know the evidence.
"This is about your character, your judgement and the warnings you ignored."
Mr Cameron appointed Coulson as his official spokesman after becoming prime minister in 2010, only for the former News of the World editor to resign in 2011 amid growing allegations about phone hacking at the paper.
The prime minister had become "the first ever occupant of his office who brought a criminal into the heart of Downing Street", Mr Miliband added.
Phone-hacking verdicts in full:
Clive Goodman - the News of the World's former royal editor - and private investigator Glenn Mulcaire have already served jail terms.
In 2007 they were convicted after hacking the phones of members of the royal household.
Royals, celebrities and victims of crime had their phones hacked by the now-defunct tabloid, which closed in July 2011 after revelations about murdered schoolgirl Milly Dowler's voicemails being accessed came to light.
Police say thousands of people's phones were targeted.
Other victims of hacking, who were not mentioned during the trial, included the Countess of Wessex and the Duchess of York.
In 2006, when hacking at the tabloid was first exposed, the police established that 19 members of the Royal Household or their close associates had their messages listened to illegally.
In the wake of the verdicts, Milly's sister Gemma urged the prime minister to deliver on his pledge to make "real permanent change" to the way the press is regulated.
Speaking after the jury was discharged, Assistant Commissioner Cressida Dick said the Met had been "conscious of the sensitivities and legal complexities of investigating a national newspaper".
The force had been criticised for its relationship with News International and for not pursuing allegations of hacking in 2006.
But Ms Dick - who oversaw the investigation - said: "Throughout the investigation we have done our best to follow the evidence, without fear or favour.
"This investigation has never been about an attack on press freedom but one to establish whether any criminal offences had been committed, to establish who was responsible for committing them and to bring them to justice. The victims deserved no less."
Coulson now faces a maximum of two years in prison for hacking.
He and Mrs Brooks had an on-off affair for at least six years from the late 1990s, details of which were revealed during the trial.
Mrs Brooks was accused of retrieving boxes - with the help of her assistant Ms Carter - from the News International archive relating to her time as editor of the News of the World and the Sun.
She was also accused, along with her husband and Mr Hanna, of hiding personal computers from the police.
Five other people have already pleaded guilty to related charges of conspiracy to hack phones.
Mulcaire, a private investigator, former news editors Greg Miskiw and James Weatherup and reporters Dan Evans and Neville Thurlbeck had previously admitted their role in the plot to get stories by intercepting voicemails.
While some reports have put the cost of the trial at up to £100m, HM Courts and Tribunal Service said it was impossible to calculate as it did not keep case by case records.
The Crown Prosecution Service said its costs alone were £1.7m - of which £1.2m was spend on counsel's fees.
A jury has failed to reach verdicts on two outstanding charges against former News of the World editor Andy Coulson and journalist Clive Goodman.