Elliott, who trains horses at Cullentra stables in County Meath, Ireland, has won a slew of major titles including two Grand Nationals -- regarded as the most grueling and spectacular steeplechase in the world.
"Firstly, I apologise profoundly for any offence that this photo has caused and can categorically state that the welfare of each and every horse under my care is paramount and has been central to the success that we have enjoyed here at Cullentra," wrote Elliott as he put out a statement on Twitter on Sunday after the image went viral online.
"The photo in question was taken some time ago and occurred after a horse had died of an apparent heart attack on the gallops. I appreciate that an initial viewing of this photo suggests it is a callous and staged photo but nothing could be further from the truth."
The image shows Elliott astride the horse, which is lying on its side. He is making a peace sign with one hand and holding his phone to his face with the other, and a caption across the photo reads: "New work rider this morning."
In his statement Elliott said he had received a phone call as he was standing over the horse, waiting to help with the removal of the body.
When the call came in he sat down "without thinking" and then responded to a shout from a member of his team by gesturing "to wait until I was finished," said the statement.
"Such background information may seem trivial at this time and will not allay the concerns of many people both within and outside the world of horse racing," continued Elliott. "However, I feel it is important to provide people with some context surrounding this photo."
Elliott ended the statement by saying he was cooperating fully with an investigation announced by the Irish Horseracing Regulatory Board in a brief statement on Twitter on Saturday.
The British Horseracing Authority (BHA) said it had been in contact with the IHRB and welcomed its investigation.
"We hope the Irish authorities will quickly confirm how this shocking picture originated," a BHA spokesperson said in a statement published Sunday.
"Respect for horses is a fundamental value of our sport, contrary to the impression in this picture," the spokesperson added. "The IHRB have assured us that the investigation will be carried out as quickly as possible and that they will keep us informed as more information becomes available."
On Monday betting company Betfair said Elliott will no longer serve as a brand ambassador for the firm.
"While we recognise that Gordon deeply regrets and apologised unreservedly for his poor judgement his actions are completely at odds with the values of the Betfair brand and that of our employees," a Betfair spokesperson said. "With that in mind, we have decided to discontinue our association with Gordon with immediate effect."
Elliott is a former jockey who started training horses in 2006, according to his website. He won the Grand National in 2018 and 2019 with Tiger Roll, which is owned by Michael O'Leary, owner of airline Ryanair.
According to the Animal Aid charity, 186 horses died in 2019 "as a direct result of racing" in Britain. The charity, which describes itself as one of the UK's largest animal rights groups, said 145 of those 186 horses were killed in jump racing.