Apart from two assists in the 2-0 Group D victory against co-hosts Ukraine, Benzema has been a disappointment and was ineffective in Tuesday's 2-0 defeat by already eliminated Sweden as a below-par France reached the quarter-finals as group runners-up behind England.
They will face world and European champions Spain in the last eight in Donetsk on Saturday.
Coach Laurent Blanc must have had high hopes that Benzema would reproduce the form he showed during a fine season with Spanish champions Real Madrid, when he netted 21 league goals, seven in the Champions League and three in the King's Cup.
However, Blanc has the 24-year-old dropping deeper to collect the ball than he is used to and opponents have found it relatively easy to thwart his attempts to create space to shoot or link up with playmakers Franck Ribery and Samir Nasri.
Tuesday's game was a case in point as Benzema made several probing runs at the bank of yellow Swedish shirts in front of him but either failed to test goalkeeper Andreas Isaksson, missed the target or had his attempts blocked.
He saw plenty of the ball and his work rate could not be faulted but it just did not happen for him against a resolute Sweden defence.
The contrast with Zlatan Ibrahimovic was striking.
The inconsistent Sweden captain fired his side ahead with a stunning volley in the 54th minute in Kiev, producing the goods a little too late with the Scandinavians already out of contention.
However, he became the first player to score at least twice at three separate European Championships having also netted in the opening defeat by Ukraine.
Asked about Benzema, Blanc told a news conference: "Like all goalscorers he needs to score goals.
"He wants to, he might be frustrated because he can't find the back of net.
"He tried a lot and could have scored but did not, like the other players. I hope he gets off the mark in the quarter-final against Spain."
Benzema was left out of France's 2010 World Cup squad at a time when he was struggling to settle at Real following a high-profile move from Olympique Lyon in 2009.
However, Mourinho's cat comparison appeared to sting him into life and he slimmed down and sharpened up to move ahead of Argentina's Gonzalo Higuain in the pecking order for the centre forward position.
A stint at a high-performance clinic in Italy helped him get into shape and a leaner, meaner version of Benzema caused havoc among La Liga defences and fed hungrily off the service of team mates Cristiano Ronaldo, Mesut Ozil, Kaka and Angel Di Maria.
Perhaps the pressure of a nation's expectations is proving too much for him at Euro 2012 or international defenders are just that much better than those he faces in La Liga.
Whatever the case, France will need Benzema to spark into life if they are to repeat their Euro triumphs of 1984 and 2000.
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