The 23-year-old spent the second half of last season on loan with the Scottish giants and impressed under the tutelage of Brendan Rodgers, scoring nine goals in seven starts and 12 substitute outings.
Idah helped Celtic retain the Scottish Premiership title and scored the winner against Old Firm rivals Rangers in the Scottish Cup final.
The Republic of Ireland international has since returned to parent club Norwich where he was allegedly facing disciplinary proceedings after he failed to turn up for the Canaries' flight to their pre-season training camp in Austria.
However, Idah was named in the matchday squad for Norwich's opening match of the new Championship season last weekend, coming on as a second-half substitute in a 2-0 defeat to Oxford United.
Idah set to Celtic medical ahead of permanent move
Idah is understood to have made it clear that he is keen to join Celtic, and negotiations between the Glaswegian giants and Norwich have been ongoing throughout the summer.
According to Sky Sports News, Celtic had an initial bid of £4m rejected by Norwich in July, but an improved offer, believed to be in excess of £6m plus add-ons, resulted in the two clubs resuming negotiations last week.
The report adds that the Hoops have now agreed a £9.5m deal with Norwich to sign Idah, including and initial fee of £8.5m plus £1m in add ons.
Norwich are also said to have inserted a 15% sell-on clause into the deal for Idah, who is ready to agree terms on a long-term Celtic contract.
Idah is expected to undergo a medical in the coming days ahead of finalising his permanent switch to Parkhead.
Idah rated highly by Celtic boss Rodgers
Idah's expected return to Celtic comes as a major boost for Rodgers, who has described the striker as a "big-game player" who had an "immense" impact at the club last season.
Speaking in May about the prospect of re-signing Idah, Rodgers told reporters: "The board know how I feel on Adam, that was the reason for bringing him in. He's been a real catalyst for us.
"I knew what I was getting because I'd seen him in the Premier League at 18. He has all the attributes. He's fast, dynamic, strong and I felt with better positioning he'd get goals.
"He's been absolutely immense for us and he's clearly someone I'd like to do something with. I've seen development in him and that's what the game's about - players improving. I just know with Adam that he's got such a high ceiling.
"He's been a brilliant team player for us. He's such a good guy, the players love him. And he's a big-game player. In so many big moments, he's stepped up to the plate for us".
Idah could soon become Celtic's fourth signing of the summer transfer window after Kasper Schmeichel, Viljami Sinisalo and Paulo Bernardo, the latter has made his loan move from last season permanent.