The singer has sued over reports naming him as a suspected sex offender.
He says his right to respect for private life was infringed and wants "very substantial" damages.
Lawyers representing all sides told a judge on Friday that parties had agreed to a one-month ceasefire so negotiations could take place.
Mr Justice Mann, who has been overseeing the latest in a series of preliminary hearings at the High Court in London, indicated that he would review the position in the near future.
The singer has taken legal action against the BBC, and South Yorkshire Police, over coverage of a raid at his apartment in Sunningdale, Berkshire, in August 2014.
His lawyers say he suffered 'profound and long-lasting'' harm and should get damages.
BBC editors have said they will 'defend ourselves vigorously'.
A spokeswoman said the BBC had reported Sir Cliff's 'full denial of the allegations at every stage'.
South Yorkshire Police have apologised 'wholeheartedly for the additional anxiety caused' by the force's 'initial handling of the media interest' in its investigation into the singer.
Lawyers say in late 2013 a man made an allegation to the Metropolitan Police, saying he had been sexually assaulted by Sir Cliff at Sheffield United's Bramall Lane football stadium when a child in 1985.
Metropolitan Police officers passed the allegation to South Yorkshire Police in July 2014.
Sir Cliff denied the allegation 'as soon as it was brought to his attention' and in June 2016 prosecutors announced that he would face no charges.