The former Downing Street spin doctor said he did not know anything about the resulting "fairly unremarkable story".
Coulson told jurors he did not remember it, or recall speaking to then editor Rebekah Brooks about it.
The 46-year-old was at the helm of the Sunday paper in April 2002 when private investigator Glenn Mulcaire hacked Milly's voicemails.
A message left on the murdered schoolgirl's phone suggested she may have applied for work in a factory in Telford.
Coulson, who was deputy editor at the time, said he remembered a conversation about the theory but said: "I thought it was nonsense because Milly Dowler was a 13-year-old schoolgirl."
On April 14, 2002, the tabloid ran a story entitled Milly Hoax Riddle.
He said he would not have known from the content that anybody at the NotW had been involved in hacking Milly's phone.
Coulson, of Charing, Kent, is charged with conspiring to hack phones with Brooks and Kuttner and conspiring with former royal editor Clive Goodman to commit misconduct in a public office.
All seven defendants deny charges against them.
The trial continues.