The 41-year-old was named as Stephen Kenny’s right hand man in April.
Duff joined Celtic’s coaching staff in February 2019 when Neil Lennon was appointed as the Hoops’ manager after Brendan Rodgers left for Leicester City.
Duff helped Lennon guide Celtic to the Scottish league title last season while they retained the honour this campaign when the 2019/2020 season was ended early last week because of the Coronavirus pandemic.
"I spoke to Damien on Monday and he was in good spirits but he'll join the FAI," said Lennon. "It’s a blow because he contributed immensely and personally I got on great with him.
"He's in his formative years as a coach but he made a huge contribution and there was a good chemistry there. So he’ll be difficult to replace, that’s for sure.
"We were waiting on everything getting called first then we’ll look to replace him but it’s not at the forefront of our minds."
Duff spent three seasons at Chelsea as a player and was part of the sides that won two Premier League titles under Jose Mourinho in the 2004/2005 and 2005/2006 seasons.
Duff also played for the likes of Blackburn Rovers, Newcastle United and Fulham, and won 100 caps for the Republic of Ireland national team, scoring eight goals.
