The 22-year-old became a transfer target of manager Chris Hughton, following a brilliant campaign last season, during which he scored 12 goals in 18 games. However, an initial bid of €8 million for him was rejected last week.
Zurich's president Ancillo Canepa had insisted they wanted to build a formidable side around the Ghanaian.
However, the Premier League new boys have made an improved offer which has been accepted. Dwamena is likely to undergo his medical within the week, according to Goal's sources.
Dwamena joined Zurich from Red Bull Salzburg after failing to command a starting role and will become the fifth Ghanaian in the Premier League after the Ayew brothers, Daniel Amartey and Christian Atsu.
Dwamena made his debut for Ghana in June when he netted twice for the Black Stars in a 5-0 thrashing of Ethiopia in the 2019 Africa Cup of Nations qualifiers at the Baba Yara Sports Stadium in Kumasi.