The Toffees' haul of 13 points from their opening five games under Koeman means that the Merseyside club have made their best start to a season since 1978.
Jagielka believes that the competition for places under a new manager is driving the players to give their best on the pitch which is in turn leading to positive results.
"Yes there is a fear, definitely," said Jagielka.
"It is like being back at school with a new head-teacher who is trying to be a little bit stricter.
"When a new manager comes in everyone has to be on their toes, there is competition for places and even the situation with the financial backing we have now adds to it all.
"Not one player is guaranteed a place in the squad let alone the team. If the manager wants to go and buy a player in January, then the cheque book is open.
"It does put added pressure on the players - but it is a good pressure."