Everton suffered a 2-0 home defeat to Manchester United.
Everton drew with Chelsea in the Premier League on Saturday.
Everton came away from Anfield with a 1-1 draw to move themselves back into the top-half of the Premier League table.
Sunderland are on the lookout for a new permanent manager.
All three candidates have either managed or played for Sunderland before.
Sunderland are still looking for a replacement for Simon Grayson, and have only just started interviewing candidates.
Koeman's future is in doubt after Everton lost to Arsenal on Sunday afternoon.
Everton were beaten 5-2 by Arsenal in their most recent Premier League fixture.
Everton's search for a striker should have led to a move for Fernando Llorente, according to Peter Reid.
Everton are currently sat in 16th place in the Premier League table.