The 23-year-old posted the photo of herself and two team-mates with the Manchester United boss at a restaurant on her Instagram account just hours after United's 4-2 win at Old Trafford.
Duggan, who joined City from Everton in 2013 and has also made 16 appearances for England, was subjected to abuse on social media after her posting, which was subsequently deleted.
Duggan said: " I would like to apologise to all MCFC fans for the error of judgement I made last night
"I now appreciate the significance of my actions and can assure everyone connected with Manchester City that I am proud to represent the Club and wear the shirt."
Duggan and her team-mates were out celebrating City's 3-1 win over Birmingham City Ladies in an FA Women's Cup quarter-final which was also played on Sunday.
Source : PA
Source: PA