The apology comes after Hales was pictured in the Sun dressed as black rapper Tupac Shakur during a fancy dress party in 2009.
"Tupacc has always been my favourite musician, but it was disrespectful, foolish and reckless on my behalf,” he said.
"I want to apologies for the offence caused and embarrassment caused to the club.
"My 20s was full of reckless off-the-field mistakes, which let down family, team-mates, friends and close relationships I had.
“Some of those decisions I will regret for the rest of my life.
“I have been away from the spotlight a bit in the last few years and have strived to better myself off the field and as a person.
"It is something I feel I have done and am continuing to strive to do.”
Hales had previously come under the scrutiny after it was suggested he gave his dog a name with racial connotations.
It led to Notts launching an investigation, with the club now issuing a further statement confirmed that they had deepened the scope of the ongoing investigation.
The statement said: “Nottinghamshire County Cricket Club have extended the scope of their investigation into the historic conduct of Alex Hales following the publication of a photograph taken in 2009.
“Alex is currently in Australia ahead of the Big Bash League and is due to return to Nottinghamshire in the new year to prepare for his commitment to the club in the T20 Blast.
“Alex will be subject to the club’s established disciplinary process and has indicated his willingness to participate in the investigation.
“He has posted a further personal statement and public apology via his Instagram account.”