The Dutch champions have been pursuing midfielder Klaassen since July, but Werder showed little interest in a sale.
However, financial problems at the Bundesliga side have opened the door for Ajax to re-sign Klaassen.
27-year-old Klaassen is under contract at Werder until 2022 and Ajax are reportedly willing to pay his full market value (€12m).
Klaassen, who was Werder’s record signing when he joined from Everton in 2018, wants to return home.
Werder barely escaped relegation last season and their lack of progress in recent years has influenced Klaassen’s decision.
The Bremen club need to plug a €30m shortfall and Milot Rashica, who has been expected to leave, remains at the club.
Selling vice-captain Klaassen would at least alleviate the financial problems at Werder.
The 16-time Dutch international made 180 appearances for Ajax between 2011 and 2017, scoring 55 and assisting 38 goals.