The former Tottenham and Paris Saint-Germain defender became a free agent after leaving Villarreal at the end of last season.
Cote d'Ivoire captain Aurier racked up 77 Premier League appearances for Spurs, also reaching the Champions League final in 2019.
Djibril Sidibe was reported to be a target of Cooper to provide competition for Neco Williams, who arrived from Liverpool in July, but Forest have instead signed Aurier, subject to international clearance.
Forest is yet to confirm the terms of Aurier's contract, with the full-back providing more cover on the right-hand side after Cooper's side failed to re-sign loanee Djed Spence, who joined Spurs.
Aurier joins the likes of Dean Henderson, Jesse Lingard, Moussa Niakhate, Morgan Gibbs-White and Emmanuel Dennis as Forest's squad reshaping continues past the end of the transfer window.
Forest sits 19th in the infant Premier League table and will look to recover from a 3-2 defeat to Bournemouth when it visits Leeds United on Tuesday (AEST).