Glasner, 51, will replace Vitor Pereira, whose exit from the City Ground came as a shock earlier this week.
Nottingham Forest's offer makes the Austrian the highest-paid manager in the club's history and one of the highest-paid managers in the Premier League, says The Sun.
Glasner left Crystal Palace at the end of last season after running down his contract and turning down offers to stay at Selhurst Park.
He was the most successful manager in Palace's history but clashed with chairman Steve Parish over the sale of Marc Guehi and his feeling that the club had dragged their feet on incomings.
Under the former Eintracht Frankfurt boss, Palace won the FA Cup, the Community Shield and the UEFA Conference League.
Glasner was also considered by both Chelsea and Manchester United after announcing his decision to leave Palace, and held talks with AC Milan and Bayer Leverkusen.
His move to Forest is likely to increase tensions between the two Premier League clubs.
Palace were demoted from the Europa League to the Conference League after Forest protested their breach of multi-club ownership rules, with former Palace co-owner John Textor also holding a stake in Lyon, who had also qualified for the Europa League.
Forest took Palace's place in Europe's second-tier competition and reached the semi-final, where they were knocked out by eventual winners Aston Villa.