Compensation between Ipswich and O'Neil's current club Strasbourg is still being finalised, but is not expected to derail the move.
Tim Jenkins and Neil Critchley, who worked alongside O'Neil at Strasbourg, are also expected to follow him to Suffolk as part of his backroom staff.
Former Manchester United manager Ole Gunnar Solskjaer had also been under consideration for the role this week, but O'Neil — previously in charge of Bournemouth and Wolverhampton Wanderers — has long been admired by Ipswich's hierarchy, with the club's interest first reported earlier this month.
O'Neil led Strasbourg to eighth in Ligue 1 last season and guided the French club to the semi-finals of the Europa Conference League, losing to Rayo Vallecano. It marked the first time Strasbourg had reached the last four of a European competition.
O'Neil previously played for Bristol City, where current Ipswich chief executive Mark Ashton held the same role he now occupies at Portman Road.
Strasbourg had been confident of keeping O'Neil after his arrival in January, but he is now set to return to the Premier League for the first time since leaving Wolves in December 2024.
McKenna stepped down last week despite leading Ipswich back to the top flight, finishing second in the Championship last season. He had taken charge of the club in 2021 and delivered three promotions across four seasons, two of which took Ipswich into the Premier League.
McKenna had been linked with the vacant Fulham job following Marco Silva's departure, but instead chose to step away from football altogether to spend more time with his family.
"I feel this is the right time for me to step aside," McKenna said. "I do so with great pride at the incredible progress we have made and with huge hope and optimism for the future of the club."
Ipswich open their Premier League campaign at home to Sunderland on 22 August, leaving O'Neil a tight window to prepare his new squad before the season begins.