Mourinho and his struggling Chelsea side have managed just eight points in as many games so far this season and face an uphill battle to recover their season and muster some kind of a title challenge.
Koeman, though, has highlighted what he believes is to blame for Chelsea's lack of form and that they should look towards their London rivals as inspiration.
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According to the Mirror, he said: "Chelsea prefer to defend when attacking is also an option.
"Even if his team goes up 1-0, they would rather defend their lead than try to score more goals.
"Yet Arsenal can kill an opponent, and this is why their victory against Manchester United was so impressive. Attacking-wise, they completely destroyed United.
"For 20 minutes, they played the most fantastic football, while Chelsea do the opposite thing."
Chelsea's defensive struggles is epitomised by the fact that they have conceded an unprecedented 17 goals in their eight league games, resulting in just two wins, two draws and four losses.
And following their 3-1 loss to Southampton at Stamford Bridge some two weeks' ago - Mourinho's third loss at the Bridge, his second this season - Koeman scrutinised the Portuguese's decision to blame the referees for their defeat, thereby undermining what was a fantastic performance and victory for the Saints.
"It was a shame that Mourinho was putting attention on the referee after we beat them.
"He wanted to have a penalty in the first half. But, in the first half at Stamford Bridge, Southampton should have had two penalties."
How Mourinho will react to Koeman's stern criticism is anyone's guess, though there's little doubt that the 52-year-old won't like to be compared to his French rival, especially given that Wenger has managed just a solitary win against Mourinho in 14 attempts.
Chelsea face Aston Villa this Saturday followed by a Champions League clash against Dynamo Kyiv, with anything but two wins likely to pile further pressure on Mourinho as manager.
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