Milan have slipped to sixth place in Serie A, three points outside the top four where they need to finish to qualify for the Champions League next season, after a miserable December.
Their last four Serie A matches have produced three goalless draws and a 1-0 defeat to Fiorentina while they were also knocked out of the Europa League following a 3-1 defeat to Olympiakos Piraeus.
"Gattuso is our coach and there is not and never has been an alternative plan," former Brazil international Leonardo told reporters after Gattuso's news conference.
"We have never contacted the coaches whose names have been mentioned during this period," he added, refereeing to former Arsenal coach Arsene Wenger and former Italy coach Antonio Conte. "We are still on course for our target, which is fourth place."
Gattuso, a fiery former midfielder who has been in charge for just over one year, said he felt he still had the club's backing.
"I speak with [club director] Paolo Maldini and Leonardo every day. There is great honesty amongst us. I feel close to them, nobody has ever told me that I have an ultimatum," he said.
"Since July 1, I've been hearing that 'Gattuso must go,' but at the moment I feel I have the backing of the directors.
"Of course, we need to get some results, otherwise the one who pays is the coach."
Milan, who have been European champions seven times but have not qualified for the Champions League since 2013-14, host lowly SPAL on Saturday as the championship reaches the halfway mark.
"The players need to focus on the pitch, not on my future. I want to continue working to pursue a path started a year ago with the boys," said Gattuso. "We are going through a negative moment but I am not thinking of my future."