The Sun on Sunday claimed that Giroud spent a night in a hotel with lingerie model Celia Kay prior to Arsenal's 2-0 home win over Crystal Palace in the Premier League earlier this month.
The article was accompanied by a photograph, reportedly taken by Kay, which apparently showed Giroud in his underpants in a hotel room.
"I apologise to my wife, family and friends and my manager, team-mates and Arsenal fans," Giroud wrote on his Twitter account.
"I now have to fight for my family and for my club and obtain their forgiveness
Nothing else matters at the moment."
Giroud, 27, is Arsenal's first-choice centre-forward and is also a key player for the France national side.
He was named on the substitutes bench for Arsenal's FA Cup fifth-round tie at home to Liverpool on Sunday.
Source : AFP