"If the offer is right," from Ajax, "Davy will leave us," Bremen sports director Frank Baumann said after Sunday's (AEST) 1-0 win over Arminia Bielefeld.
"There were one or two (other) offers, but they were not to our satisfaction."
The transfer window closes in Germany on Tuesday (AEST), a day before it shuts in the Netherlands.
The 27-year-old Dutch international joined Bremen from Everton in 2018 as the German club's record transfer at a cost of around €13.5 million ($22.3 million).
He has scored 16 goals, created 13 more in 81 appearances for the north German side.
He captained the team on Sunday (AEST) and helped Bremen avoid relegation last season, only after winning a two-legged play-off.
Klaassen is a product of Ajax's academy.
He made his first-team debut in 2011, winning three Dutch league titles and making 180 appearances and scoring 55 goals before joining Everton in 2017.