News International's offices in Wapping are also currently being searched by officers as part of the investigation.
Two men, aged 48 and 56, were detained at their homes in Essex on Saturday morning (January 28).
A third man, also aged 48, was arrested at home in North London, while a 42-year-old man was detained later on Saturday by appointment at a London police station.
News Corporation has since confirmed that the men are all current or former Sun employees.
A 29-year-old police officer - who is serving with the Metropolitan Police's Territorial Policing Command - was arrested at his place of work.
"The arrests were made between 06.00 and 08.00 by officers from Operation Elveden, the MPS [Metropolitan Police Service] investigation into allegations of inappropriate payments to police," a statement from the force read.
"The home addresses of those arrested are currently being searched, and officers are also carrying out a number of searches at the offices of News International in Wapping, east London. These searches are expected to conclude this afternoon."
The statement - issued before the fifth arrest - added: "Today's operation is the result of information provided to police by News Corporation's management and standards committee. It relates to suspected payments to police officers and is not about seeking journalists to reveal confidential sources in relation to information that has been obtained legitimately."
A total of 13 people have now been arrested in conjunction with Operation Elveden.
The Metropolitan Police is also conducting Operation Weeting, which is the investigation into allegations of phone hacking by the press.
A judicial inquiry into press standards, chaired by Lord Justice Leveson, is also ongoing.