Burnley, who did not sign Gray until last summer when he moved for £6 million from Brentford, do not condone prejudiced behaviour.
The striker later tweeted: "Firstly, I want to offer a sincere and unreserved apology to anybody I may have offended in relation to these tweets. The tweets were posted four years ago when I was a completely different person to the man I am now. I was at a very different point in my life back then - one that I've worked hard to move on from.
"Thankfully a lot has changed in my life since then. I have experienced a lot over the past four years and have had to take responsibility for a number of things in my life which has enabled me to mature and grow as a person. I've worked incredibly hard to completely transform my life since that time.
"To clarify, I do not hold the beliefs written in those tweets whatsoever. I can assure everybody that I am absolutely not homophobic and I can only apologise and ask for forgiveness to anyone I offended."
Openly gay rugby union referee Nigel Owens sent Gray a message in response to the circulation of the tweets. "It's your comments that makes decent human beings sick," said Owens, who received online abuse after announcing he was gay. "No wonder no one can be themselves in soccer."
Other derogatory tweets from Gray also emerged, one referring to a fatality on the motorway. It read: "Fuming. Stuck in traffic because some sad weak fool decided to get themselves run over. #NoSympathy."
