Barber, Kay and Brooks face one count of conspiracy to commit misconduct in a public office between January 1 2004 and January 31 2012.
Operation Elveden is being run alongside two others: Operation Weeting, which is looking at allegations of phone hacking, and Operation Tuleta, an inquiry into accusations of computer hacking and other privacy breaches.
Eight people including Coulson and Brooks face charges linked to an alleged conspiracy to hack phones.
The others are private investigator Glenn Mulcaire and five former News of the World journalists - ex-managing editor Stuart Kuttner, former news editor Greg Miskiw, former head of news Ian Edmondson, ex-chief reporter Neville Thurlbeck and former reporter James Weatherup.
They are all due to face trial in September next year.
So far 18 people have been arrested as part of Operation Tuleta, but no-one has yet been charged.
There is a remaining suspect who is still being investigated in relation to the charges faced by Brooks, Kay and Barber.
Prosecutors said the file passed to them by police related to two journalists and two public officials.