Celtic, champions in 13 of the past 14 seasons, sit second in the Scottish Premiership, six points behind Hearts and ahead of Rangers on goals scored.
Their points total of 38 from 20 games is 15 down from the same stage of the previous campaign.
Under Nancy, Celtic shipped 18 goals, one more than they had conceded during the opening 24 games of the season.
“Nancy was fixated on changing Celtic’s style, going to 3-4-3, playing in one particular way,” the club’s former striker Chris Sutton told Sky Sports.
“He just wasn’t adaptable. For Celtic to lose six out of eight games and ship as many goals as they did, it was kamikaze stuff.”
O’Neill, who managed the club from 2000-05, had been placed in temporary charge after Brendan Rodgers, who was appointed manager of Saudi Pro League side Al-Qadsiah last month, resigned in October.
Having previously managed rivals Montreal, Nancy led Columbus to the MLS Cup in 2023 and then the Leagues Cup – a competition between Mexican and American sides, as they finished second in the league.
However, after being named MLS manager of the year for 2024, his side slipped to seventh in Eastern Conference and 12th overall last year.
As he leaves Celtic, Nancy has enjoyed just five wins in his latest 20 games as a coach.
Serving from June 1999 to February 2000, John Barnes had previously been Celtic’s shortest serving permanent manager, overseeing 29 matches in his eight months.