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The team's manager is under pressure after four losses in five matches, despite a strong start to the season which sees them at the top of the Premiership.
Ex-Rangers player, Phil Bardsley, who O'Neill managed at Sunderland, told the Daily Record that speculating about the veteran's replacement is disrespectful.
He said: "I just think it's disrespectful to Martin O'Neill. I'm not quite sure what's going on."
Bardsley praised O'Neill's accomplishments, stating that his work last year was incredible, especially at a time when Hearts and Rangers seemed to have the upper hand. He said: "For Celtic to nip it at the end was an incredible job from Martin and his staff."
Bardsley acknowledged the team's tough start and off-field issues, including ownership problems and a strained relationship with the fans, but believes these should not overshadow O'Neill's contributions.
He said: "It's totally disrespectful to the manager, and I wouldn't be best pleased if I were Martin O'Neill."
Celtic's recent poor form has put O'Neill's role under scrutiny after a promising start to the season. The team has suffered defeats against Rangers, Ferencvaros, and Benfica, although they managed a 2-0 victory over Motherwell.
Their most recent loss was a 4-1 defeat in Lisbon to Benfica in the Europa League.