The Tigers' 1-1 draw with Millwall on Tuesday night was their fifth game in succession without victory, while the club have only won two of their last 13 fixtures.
The Hull Daily Mail claims that the team's board are already searching for replacements for the Russian, but Slutsky has insisted that he is not worried about his future.
"I am not worried about my job because I can't work if I am unhappy or in discomfort but now we have a really good atmosphere," the 46-year-old told reporters.
"Everyone supports me. That starts from the owner and finishes with each guy from the staff."
Hull sit four points above the relegation zone in the second tier after 18 games this term.