The 40-year-old has made an indifferent start to his reign, with Liverpool already knocked out of the FA Cup and League Cup.
Despite promising performances against Arsenal, Chelsea and both Manchester clubs, the Reds are 12 points behind fourth-place Spurs in the Premier League.
But Grobbelaar believes Rodgers’ first season will be deemed a success if there is a clear signs of progression and his players produce “good football”.
“If you look at the owners of the club they are not very gung-ho about winning trophies,” he told TalkSport. “They want to try and get things right.
“They are a more traditional style of owners. I don’t think they need to win a trophy this season, as long as they can progress and play good football for the supporters [that will be success].
“They are so Jekyll and Hyde. The supporters must be tearing their hair out, they are brilliant and the next game they are hopeless.”
The former Zimbabwe goalkeeper, who won six top-flight titles at Liverpool, also warned Rodgers that he cannot rely on his young players to transform the club’s fortunes.
“Brendan Rodgers cannot rely on the youngsters all the time,” he added. “He’s got to use the older stalwarts. The young guys haven’t got the stamina.
“Raheem Sterling, Jonjo Shelvey, even Joe Allen, if they can just give them a rest and bring on some of the older players, I’m sure that they will turn the season around.”
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