Forest were linked with a move for fellow Serie A centre-back Roger Ibanez from Roma last week, but the Brazilian has opted for a move to Al Ahli.
The Daily Mail claims that Forest are set to offer around £17m for the Turkish international, as defensive numbers remain light ahead of the start of the season.
Demiral has fallen down the pecking order in Bergamo over the past 12 months, since making his loan move from Juventus permanent.
One Italy's finest young prospects Giorgio Scalvini enjoyed a breakout season during 2022-23, and in doing so, took Demiral's spot in the side.
La Dea also have the experienced Rafael Toloi and Berat Djimsiti, who are preferred by manager Gian Piero Gasperini, as well as Jose Luis Palomino in reserve.
With the European Championships on the horizon next summer, Demiral may need to move elsewhere in order to keep his place in Stefan Kuntz's Turkey squad.
Despite that though, Forest are expected to have a battle in persuading Demiral to join.
Teams in Turkey and Serie A are also said to be interested, with Inter Milan the most illustrious of the clubs.
After already adding Yann Aurel Bisseck to their ranks this summer, Inter may look for further reinforcements at centre-back, especially after losing Milan Skiriniar to Paris Saint-Germain.
Forest themselves need additional cover in that position, with Moussa Niakhate set to miss the start of the season.
Steve Cooper was able to welcome back Willy Boly for their 0-0 draw with Eintracht Frankfurt on Saturday to play alongside Scott McKenna and Joe Worrall, but outside of that, Forest have no other senior central defenders.
A 5-0 thrashing at the hands of Rennes suggests additional quality may need to be added too.
Matt Turner is set to be the man to finally fill the void left by returning loanees Keylor Navas and Dean Henderson in goal, with the Arsenal man close to joining for a fee which could rise to £10m.
A left-back is also said to be high on Forest's wishlist, as new man Ola Aina has been forced to play on that side throughout pre-season after Renan Lodi preferred a move to Marseille instead of making his loan move permanent at the City Ground.