The Reds have been eliminated from the FA Cup and League Cup this season but are through to the knockout stages of the Europa League.
Grobbelaar believes that Rodgers's first season at Anfield will be deemed a success if there are signs of progress on the pitch.
"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."
Liverpool face Zenit St Petersburg in the first leg of their Europa League last-32 clash this evening.