Earlier this week reports emerged that a group of fans sent a letter to club owners Venky's asking them to consider his position after losing their first three games of the season.
Kean travelled to India last week to speak to the owners and on Sunday secured his first point of the campaign, drawing 1-1 with Fulham at Craven Cottage.
Kean admitted he was pleased to come away from his former club with a point and believes it indicates the future is bright.
"Speculation in the press has just been speculation in the press, not from within the club," he said.
"We should have won the game against Everton, and the way we played here showed great spirit.
"I've got fantastic backing from the owners, and I'm in contact with them every day. They're happy with the way we're playing.
"We had backing in the window, and have a competitive squad, so we have a squad that will finish in the top half.
"It's been a slow start, but we should already have had three or four points on the board before this game. We'll stick at it."
Both sides came into the match on the back of miserable starts to the season and Sunday's share of the spoils did little to lift the spirits of either sets of fans.
Fulham edged first-half proceedings but fell behind to a moment of magic from Ruben Rochina, who launched an unstoppable 18-yard drive in the 32nd minute.
The home side equalised six minutes later through Bobby Zamora's own inspired moment and Fulham manager Martin Jol was disappointed that his side could not find a winner.
"It was frustrating," he said. "It was similar to the [goalless draw] against Villa, especially the second half.
"We deserved three points from that game and this time it was probably the same.
"More and more nowadays in football for example you see Liverpool (on Saturday at Stoke) with 70 or 75% possession and still lose the game so that can happen.
"It is not easy to play in their half and we did that.
"We tried to do that in the first half but couldn't because we were a bit sloppy in midfield and everybody was bombing forward.
"And of course the style of this club is to try and play attractive football and in the second half that certainly happened.
"In the second half I can remember we had eight or nine crosses to the near post.
"They protected that well, but then we need to come up with good solutions.
"We needed better finishing. I've heard the phrase that we lacked a cutting edge and maybe today (Sunday) we did."