The Leeds United owner wasn't too complimentary about Evans in a recent newspaper interview.
But speaking to BBC Radio Leeds on Monday, Evans, who is on a rolling contract at Elland Road, said he believes in his head and his heart that he will be in charge next season.
And the Scot added that should he remain at the helm, Leeds will "200 per cent" challenge for the Premier League.
"I think I’m the best person to take it forward, of course I do," Evans told BBC Radio Leeds ahead of Tuesday's visit of Wolverhampton Wanderers. "Whether the president decides that I’m not, I’ve always maintained that it’s his decision - he owns the football club. He’s entitled to make the decision he thinks is best for Leeds United.
"I may not like it, it may hurt and I may shed some tears but I have to respect it."
Asked if he was upset by Cellino's comments, of which he was informed by reporters beforehand, Evans replied: "No (he doesn’t feel upset).
"I think from what you get told and what you read sometimes you go away from being upset with the benefit of experience.
"Somebody was going to be new head coach here three weeks ago or four weeks ago and I think every week since, and it was probably happening before Christmas and I only came in in October!
"It tends to never stop here. What you find is everyone has an opinion on Leeds United."
For all the speculation, Evans became the longest-serving manager of the Italian's tenure courtesy of Saturday's win over Reading, having surpassed Neil Redfearn's tally of 33 matches.
Speaking to the Telegraph, Cellino denied reports he is set to replace Evans with Fabio Cannavaro, but claimed that Carlo Ancelotti would be open to the role should Leeds return to the Premier League.