Valencia finished fifth in the Primera Division last term, but this year they have started terribly, winning just one of their opening four games.
Los Che fans vented their anger at the Serbian tactician and his players on Thursday night when they lost 3-0 to Swansea in the Europa League.
Some supporters were so angry that they left their seats at the start of the second half and after the game there was a huge protest involving thousands of fans outside the Mestalla.
Djukic called on his team to bounce back with a win against Sevilla on Sunday and insisted that he is not worried about being axed after just three months in the post.
"I don't fear for my job, the only thing that I fear is about the poor image that we are projecting," the Serbian told a press conference.
"We have to reset our heads now and start over again."
Djukic took responsibility for the poor showing against the Welsh side, who had one more player for 76 minutes following the dismissal of Adil Rami for a foul on Wilfried Bony.
Djukic thinks that, after so many years of success in the past, the pressure is starting to get to his squad.
"They are players of such great quality, but they can't put three passes together, that shows how much pressure they are under," Djukic added.
"It was a bad performance - we're not sure at the back nor are we solid up front - but we have to keep working.
"We can understand why the fans are angry because we haven't given them anything.
"At the moment it's not working, but I don't have a magic wand to fix things so we'll just have to keep working."
If Djukic thought his season had started badly, then he should take a look at his opposite number Unai Emery.
The Sevilla coach led his team to ninth last season, but the Andalusian team are second from bottom this term following their winless start in the league.
But unlike Valencia, Sevilla go in to the match on the back of a victory - albeit in the form of a Europa League triumph over the unheralded Portuguese side Estoril.
Sevilla goalkeeper Javi Varas thinks facing Valencia on Sunday at the Mestalla could prove tricky despite the hosts' current form.
"It's a bit of a double-edged sword," the 31-year-old said.
"They are going through a bad spell and they want to turn the situation around. We also want to get out of this painful run so it's a complicated match."
The good news for Sevilla is that Real Madrid loanee Denis Cheryshev trained with his team-mates on Friday despite his recent injury problems.
Emery will give Diego Perotti a fitness test before deciding whether to include the midfielder in his squad.