Rooney was thrown into the spotlight after The Sun tabloid alleged the Manchester United forward drunkenly gatecrashed a wedding at England's hotel in Watford, north of London, on Saturday.
Media reports said up to 10 other England players who separately went to a London nightclub have also been spoken to.
Rooney started England's 3-0 World Cup qualifying win over Scotland on Friday, but suffered a knee injury that ruled him out of Tuesday's 2-2 friendly draw at home to Spain.
"Although it was a day off for the whole squad and staff, he fully recognises that the images are inappropriate for someone in his position."
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The statement said Rooney had spoken to Southgate and FA technical director Dan Ashworth "to unreservedly apologise".
"He would like to further extend that apology to any young fans who have seen these pictures."
An FA spokesperson said: "All England personnel have a responsibility to behave appropriately at all times. We will be reviewing our policy around free time whilst on international duty."
- Drank like devils -
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A Rooney spokesman said: "As he has always been, Wayne was happy to sign (autographs), pose for photos and chat with guests.
"It is sad that one or two of them have now sought to turn Wayne's friendly good nature to their advantage."
And Liverpool boss Klopp also said the episode was not serious.
"I know we're all on the sunny side of life, we earn a lot of money and do the job we love, but at the end maybe it comes as a surprise that we are also human beings too," Klopp told a team press conference.
"All the guys, all the legends we love and admire they drank like devils and smoked like crazy, but they were still good players. No one does it any more. I don't know anyone now."
England's interim manager Gareth Southgate was adamant that Rooney had not been left out of the Spain game as a disciplinary measure.
"Wayne picked up an injury in the game (against Scotland) on Friday," said Southgate. "He couldn't train on Sunday and that's it."
"Over the last few years, the players have been allowed to go home during a period of time," Southgate told reporters after England's 2-2 friendly draw with Spain at Wembley.
"They trained Saturday morning and they trained again on Sunday afternoon, and I gave the players a period of time off (in between). I'm not aware of anything else.
"There were lots of changes we made to routines. Some things we thought we should keep the same and I'll have to review (it) -- or maybe I won't have to review! Someone will have to review it.