Their poor run has seen Spalletti come under scrutiny, with reports claiming that Conte, who is yet to return to management after leaving Chelsea, had been at Inter's offices to meet with CEO Beppe Marotta.
Conte subsequently denied there had been any contact, and in a vehement defence of his position ahead of Inter's match with Bologna on Sunday, Spalletti fired back at the Italian press.
"Conte can go for a wander wherever he wants," Spalletti told a news conference.
"You said it's almost inevitable that I will be leaving at the end of the season.
"You say that Marotta is a consummate professional on the one hand and then on the other you say that he met Conte at the club's offices.
"You know exactly what you're doing there. The club's offices are in the centre of Milan where there are millions of people.
"That's bogus information because they wouldn't meet him there.
"It's only right that Marotta works in the best interests of the club. We need to get on with our job and work hard day in, day out."
| #Spalletti: "The fans must understand that we need to remain solid and united. Through their support, we succeed in overcoming some of our insecurities. #Perisic is our player, and from now until the end of the season we must get results." #InterBologna pic.twitter.com/EhQq8WGshR— Inter (@Inter_en) February 2, 2019
Spalletti is under contract at San Siro until 2021, having signed a new three-year contract following an impressive maiden season in which Inter qualified for the Champions League for the first time since 2011.
But despite acknowledging Inter's need for success, Spalletti insisted he must be given time to implement his style of play.
"We went out of the cup just as we did in the Champions League and they are both very important competitions," Spalletti added.
"However, we need to keep working in the right way, we must continue to work on the ideas that we have been building.
"The fans are in a hurry to speed up this process because Inter haven't lifted silverware in many years. I can't bear the brunt of a number of years without trophies.
"In that period Inter arguably moved further away from winning trophies. We need to consolidate what's been done well and improve what hasn't worked. I don't think we need to start again from scratch.
"I didn't ask for anything at the start of the season, the club offered me a three-year deal.
"When you pen terms for a significant period like that, it shows that you need several years to work on the project and further improve the way you work."