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Forest had previously had three bids rejected by Wolves for the England U21 international, according to Sky Sports News.
But the Premier League new boys have now succeeded with an improved offer which is set to make Gibbs-White the club's new record signing.
Forest are set to pay an initial £35m for Gibbs-White, with a further £9.5m available in potential add-ons.
It smashes the club's previous record fee of £18.45m paid to sign Taiwo Awoniyi from Union Berlin earlier this summer.
Gibbs-White will undergo a medical with Forest on Friday and is set to become the club's 16th signing since they achieved promotion back to the Premier League.
Fans on social media have been blown away by the fee agreed for Gibbs-White, who has scored three goals in his 88 appearances for Wolves.
"Wolves laughing all the way to the bank on this one!", wrote one fan on Twitter.
Another added: "Is anybody looking at where this money is coming from?! BTW, £44m for this guy. I mean, he looks very decent, but...."
A third wrote: "Forest be making moves bro they mean business."
That’s ridiculous— ryanwalker04 🇫🇮 (@ryanwalker04) August 18, 2022
Whatttttttttttt?44.5?— Splendid AFC (@splendidmoore) August 18, 2022
City paid £52m for Haaland btw— Josh (@JoshRFC_) August 18, 2022
That's unbelievable, Wolves had their pants down big time— Aaron (@AaronBates91) August 18, 2022
£44m for Morgan Gibbs-White? pic.twitter.com/nAdbLy9UtG— Footy Accumulators (@FootyAccums) August 18, 2022
While another fan commented: "City paid £52m for [Erling] Haaland btw.."
Gibbs-White is set to reunite with Forest manager Steve Cooper, who he played under while on loan at Swansea in the 2020-21 season.
He was also part of Cooper's victorious England U17 squad that won the 2017 FIFA U17 World Cup in India.