Forest lost 2-1 to Leeds United on Tuesday to extend their winless run in the top flight to eight games.
The City Ground outfit are now just outside the relegation places on goal difference.
Reports on Wednesday suggested Forest were set to sack Cooper but owner Evangelos Marinakis, despite admitting results need to improve, has given the Welshman his backing.
Marinakis said in a statement: "No one denies that our club is in a difficult position in the Premier League, but we wish to end the speculation and the false and disruptive reporting in the media to confirm that Steve Cooper remains our manager at Nottingham Forest.
"We have all been disappointed with recent performances, and it is very clear that a lot of hard work needs to be done to address this urgently. Results and performances must improve immediately.
"Now is the time for everyone connected with our club, from us as owners, to the board, our supporters, backroom staff, coaches and players, to come together and fight to secure our status in the Premier League.
"There can be no time for distractions, rumours and speculation.
"There is only time for hard work, determination, a commitment from Steve and the players to getting the results we need and, of course, the continued magnificent support of the fans of Nottingham Forest."
