The FA Cup trophy. Picture: Ash Donelon/Manchester United via Getty Images
The draw for the last eight took place before this Thursday night’s fifth-round tie at Goodison Park.
And the Toffees will travel to Selhurst Park in the quarter-finals of the competition.
Everton lost 3-1 at Crystal Palace in the Premier League earlier this season.
Meanwhile, Liverpool make the trip to either Championship pair Nottingham Forest or Huddersfield Town.
The duo meet at the City Ground on Monday night.
The Reds defeated Norwich City on Wednesday night and harbour hopes of an unprecedented quadruple.
Ties take place between 18-20 March.