The Dutchman had earned some respite in midweek when Inter beat Torino to end a three-game losing streak, but Sunday's 1-0 loss has piled the pressure back on.
"De Boer, it's finished" read a Corriere dello Sport headline after Inter, enduring a fifth season without Champions League football, dropped to 11th in Serie A.
De Boer, who had never previously coached or played in Italy, was appointed only two weeks before the start of the season after Roberto Mancini left by mutual consent.
He is Inter's eighth coach since Jose Mourinho left in 2010 after winning a Serie A, Coppa Italia and Champions League treble.
De Boer said slow starts and players failing to win their individual battles were to blame for the malaise.
"I'm happy to be here and I believe we can change this situation," the Dutchman said after the game. "The situation is difficult and I see that we perform well in fits and starts but not for a whole match.
"We improved a lot in the second half but we cannot keep starting like this. It's a question of mentality and I cannot accept it because we have quality in this team.
"I didn't see us with the character and confidence we needed to win the duels," he added.
"Lots of the players didn't play to their best. I was angry at half time."
Asked about his position as coach, he said: "You'll have to talk to the club."
Corriere dello Sport listed Brazilian Leonardo, Frenchman Laurent Blanc and Italians Francesco Guidolin and Stefano Pioli as favourites to replace De Boer should he leave.
(Writing by Brian Homewood; Editing by Peter Rutherford)
