The West Brom boss, who was Kenny Dalglish's No.2 with the Reds last season, says the Northern Irishman must lead his side back into the Champions League during his first term
Clarke was Kenny Dalglish's No.2 at the Merseyside club last season, but was sacked alongside the Scot this summer following a disappointing eighth-placed Premier League finish.
And the 48-year-old, who has since taken the top job with the Baggies - Liverpool's opponents on the first day of the new season - believes Rodgers could face the same fate.
"The owners have said top four is not imperative for the current season," he told reporters. "I don't believe that.
"A club like Liverpool have to be there challenging for the top four – I don't think they can accept anything less."
The former Chelsea coach also argued that Liverpool's League Cup victory and journey to the FA Cup final last term were proof of the progress made by Dalglish.
"When you look at the team we were starting to build and to reach two cup finals," he added. "If you are reaching cup finals, you know you have a team who can compete. It is not an easy achievement. Not many teams do it."
The former Scotland international defender went on to defend the club's transfer record under Dalglish, who spent vast amounts of money on the likes of Stewart Downing and Andy Carroll following the departure of big-name players such as Fernando Torres.
"We signed good players," he insisted. "Whether they get value for money, it depends on how they are used this season. Jordan, Andy, Stewart and Jose Enrique were all in their first seasons.
"There were a number of new signings and it takes time for them to bed in. It took Andy quite a while to settle into the club and the way we wanted to play, but towards the end he was making a big contribution. He is a really good player."
Clarke also revealed that Rodgers was open to keeping him at the club as assistant manager before owners Fenway Sports Group opted for a complete clearout.
"I actually got sacked by Liverpool," he said. "I spoke to Brendan after Kenny left and I was still there. Brendan said he was one of the candidates and told me, 'Don't do anything silly because maybe we could work together'.
"But after they took Brendan in, they wanted to re-jig and reshape the whole staff. The owners wanted to clean the place out and start with fresh people. There were two or three jobs still available and I was fortunate to get this.
"[Saturday's clash at the Hawthorns is] a big game for me because it is the first game in charge at West Bromwich Albion."
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