Koeman was appointed the Everton manager in the summer of 2016 after leaving Premier League rivals Southampton at the end of the 2015-16 season.
The Dutchman guided the Toffees to a seventh-placed finish in the Premier League last season.
Much is expected of the Merseyside this campaign, but things are not going according to plan.
Everton are 16th in the Premier League table at the moment with just seven points from as many matches, just two points above the relegation zone.
Former Arsenal star Keown, who now works as a pundit for BBC Sport and BT Sport, has stated that sacking Koeman would be premature.
The former England international, though, wants the mood at Goodison Park to change.
“It would be premature to sack him,” Keown told The Daily Mail when asked if Everton should sack Koeman.
“Do not forget that this is a manager who has been very successful and delivered Everton's highest finish since 2014.
“He has also had to contend with the departure of a valuable player with Romelu Lukaku. What needs to change is the mood at Goodison.
“I watched their disappointing draw with Apollon Limassol in the Europa League and could see the manager carrying a heavy burden of pressure.
“The mood is affecting the players. I can imagine they were relieved to escape the atmosphere this week when they went away with their national teams.”
Everton will return to action on Sunday afternoon when they take on Brighton and Hove Albion away from home at the American Express Community Stadium in the Premier League.