The Seagulls have made a solid enough start to life in the Premier League, picking up seven points from as many games, while they have won their last two home matches in the top-flight against West Brom and Newcastle.
Brighton do not have the best record against Everton, however, as they have only won two of the 11 previous meetings between the two sides, although they have not clashed in a competitive match since 1983.
Boss Chris Hughton will be able to call upon midfielder Davy Propper for Sunday's match, as he has recovered from a minor hip injury that disrupted his international break, although Tomer Hemed is banned and Sam Baldock and Beram Kayal remain sidelined with long-term injuries.
Everton will head into the match having lost four of their last five league games, with the pressure seemingly mounting on Koeman, particularly as they were amongst the biggest spenders in the top-flight during the summer.
The Toffees have particularly struggled on the road of late, as they have failed to win any of their last 11 away matches in the league since beating Crystal Palace at the start of the year.
Koeman will be boosted by the news that captain Phil Jagielka is ready to return following a hamstring problem and he could replace out-of-form Ashley Williams at the heart of the Toffees defence.