Video clips of the chanting, which were taken before a pre-season friendly at Sunderland on Saturday, July 29, have emerged.
Celtic, who are the Scottish champions, won the match 5-0, but the game was marred by fighting between Celtic and Sunderland fans.
Corporal Rigby, who was 25, was attacked with knives and a machete and almost decapitated by Daesh-supporting Michael Adebolajo, 28, and Michael Adebowale, 22, in London in December 2013.
He was attacked in the street as he returned to Woolwich barracks in south east London.
His killers claimed they were avenging Muslims killed in Afghanistan and Adebolajo described himself as a "soldier of Allah."
After their conviction, a report accused MI5 of failing to pick up on the danger the pair presented.
It later emerged that MI6 failed to respond adequately to Adebolajo's accusations of being mistreated by Kenyan counterterrorism units in 2010, before he was deported to the UK.
Corporal Rigby, who was born in Manchester, joined the British Army in 2006 and was selected to be a member of the Corps of Drums, posted to the 2nd Battalion, The Royal Regiment of Fusiliers.
The Celtic fans, who are thought to have been drunk, alluded to his profession in the army during their sick song.
WARNING: Offensive content
Corporal Rigby's mother Lyn was interviewed by a Scottish newspaper after the clip emerged on YouTube.
"I haven't stopped crying since I saw that video," she said.
"Celtic need to denounce these scum wearing their team's shirt and apologize for what they have done," added Mrs. Rigby.
"I was on the same train as them on Saturday… They truly are lowlifes. Celtic fans are always chanting for the terrorist scum. It makes me ashamed to be Scottish," wrote one user on YouTube.
Celtic, a club which traditionally drew support from Catholics who share an affinity for Ireland and were opposed to the British Army's role in Northern Ireland, put out a statement on Wednesday, August 2.
"Everyone at Celtic is appalled by these events. Such behavior in no way represents Celtic Football Club or our supporters," it said.
"We understand this incident is now subject to an ongoing police inquiry," they added.
"We're aware of the chants that were made and are looking into these to see if any criminal offences took place and will take necessary action," said a Northumbria Police spokesman, in an email to Sputnik.
Fans of Celtic and their traditional rivals, Glasgow Rangers, have a long history of singing offensive chants.
Rangers fans — who are predominantly Protestant — have been banned from singing a song called the "Billy Boys," the lyrics of which refer to being "up to our knees in Fenian blood." A Fenian is an offensive and derogatory term for a Sinn Fein supporter or a Catholic in general.
Such is the mutual hatred between the two sets of fans that when Celtic fans began flying Palestinian flags in the 1990s, Rangers supporters responded by buying Israeli flags.
Tonight folks as a video showing @celticfc fans glorifying the death of Lee Rigby let's show the family we support them. Please RT 2 support pic.twitter.com/4tiBv9o3HN
— Lee Rigby Mem Cup (@CupRigby) July 31, 2017
Likewise when Celtic fans began flying the ikurrina — a flag associated with Basque independence from Spain and the ETA separatist group — Rangers fans responded by unfurling Spanish flags.
Rangers fans will even wear England shirts to games to goad their rivals.
