Shilton believes Forest's new signings will make difference
Billy Davies's men secured their Championship survival with one game to go after a disappointing season.
But Shilton, who played 202 games for the club, including two European Cup finals, thinks new signings David McGoldrick and Dele Adebola could make the difference.
The 59-year-old, who is the most capped England player with 125 caps, said: "Forest need to start moving up because they are a big club with a great history and a massive fan base.
"Billy Davies came in as manager and kept them in the Championship but now he has had chance to bring in some of his own players.
"They need to start pushing on and with the players they have brought in they should be looking at the play-offs this year.
"It's so important for the fans that the club is up there because they do live on their past glories at the moment.
"Time will tell if Billy Davies is the right man to get the club going in the right direction.
"He has been in the game for a while and had some success but I think this is going to be a big season for him and for the club.
"The fans will demand success."
Brian Clough signed Shilton one month into the 1977 season from Stoke City for a fee of £250,000.
That year he played a key role as the club secured the Division One league title and he was honoured with the PFA Player of the Year award.
During his time at the City Ground, Shilton won two European Cups, a European Super Cup, the First Division championship and the League Cup.
Dad of two Shilton became the first footballer to play 1,000 league games when he turned out for Leyton Orient against Brighton in 1996.
He also has the distinction of playing more than 100 league games for five different clubs.
He finally retired in 1997 at the age of 47.
Shilton was speaking at the launch of the new Sony Ericsson W995 mobile phone in London. The handset is available exclusively on the '3' network and comes with three months free Sky Mobile TV.
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