Prosecutor Andrew Edis QC told the Old Bailey that the News of the World (NotW) heard a false rumour in May 2005 that Mr Clarke was seeing his "attractive special adviser", Hannah Pawlby.
The newspaper tasked private investigator Glenn Mulcaire with hacking Ms Pawlby's voicemails and "door-stepped" her, but Mr Coulson also called and left her voicemails, the court heard.
Mr Edis said: "The prosecution suggests that Mr Coulson, who is now the editor of the NotW, he is not the man who stands outside people's houses hoping to catch them out, he is the man who likes to put the story to people to see what they will say."
He said the NotW used three ways to investigate stories -phone hacking, surveillance, and confrontation.
"The editor is personally involved in the third. Obviously he knows about the second, surveillance, he must do. What about the first? Does he know about phone hacking? He says he doesn't, we say 'Oh yes, he did'."
But there was no affair between Mr Clarke and Ms Pawlby, the jury heard.
Rumours were first picked up by the NotW's features desk when a source who was sexually interested in Ms Pawlby was told: "Don't bother wasting your time - she's with Charles."
A tape of voicemails taken from her phone on at least three occasions was seized from Mulcaire's home in August 2006. Investigators also found entries on the private investigator's computer which had Ms Pawlby and her sister as "Projects".
During the period she was being investigated, Ms Pawlby's grandparents received anonymous calls asking for information about her, Mr Edis said.
Meanwhile, former chief reporter Neville Thurlbeck and former reporter James Weatherup oversaw surveillance of Ms Pawlby's movements.
Leaving her a voicemail on June 18, 2005, Coulson told her: "I've got a story that we're planning to run tomorrow that I really would like to speak to Charles about. I wouldn't do this in the normal course of events but it's quite a serious story and previously Charles has been very helpful with us, suggesting that when there are these issues that pop up we should speak directly with him.
"So if you could ask him to give me a call or, of course, give me a call yourself, I will fill you in."
A similar ploy had been used against another Labour home secretary, David Blunkett, to help expose his relationship with a woman.
Mr Edis said a hairdresser called Laura Rooney had her phone hacked, even though she had no connection with England striker Wayne.