Mangala, who is on loan from Lyon, sustained an anterior cruciate ligament knee injury during their 1-0 win at Brighton & Hove Albion on Saturday. He made 19 league appearances for Everton this season.
Calvert-Lewin had scored his first goal in 17 games in all competitions when he netted in a 3-2 victory over Tottenham Hotspur but sustained a hamstring injury in the 12th minute at Brighton and the club are seeking a specialist's opinion.
Everton, who are 16th in the standings after two straight wins under new manager David Moyes, are seven points above the relegation zone. They next host 17th-placed Leicester City in the league on Saturday.