Luciano Spalletti had been looking to make it five wins from five but had to settle for just one point in Bologna.
Verdi came out firing on all cylinders with a half dozen shots at goal before the breakthrough on 32 minutes with a powerful long-range effort.
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But Argentine forward Icardi — sporting a new shaven head — levelled from the spot at the 77-minute mark after a contested foul by Ibrahima Mbaye on Eder.
"It's not a scandal for a team to draw," said Spalletti. "We were slower than usual, unable to make the ball move well in vertical lines, especially in midfield, so we gave Bologna the chance to close down our channels."
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Verdi's was just the second goal conceded this season by Inter, who had been sitting top on maximum points alongside Napoli and Juventus after four matches.
Now with 13 points, Inter move provisionally top but champions Juventus and Napoli have the chance to pull ahead when they play Fiorentina and Lazio respectively on Wednesday.
"We knew that Bologna is a team that plays well," said Inter striker Eder. "There are two lost points, we will see what went wrong this week."
Eder added: "We have to look at ourselves not Napoli and Juventus, our goal is the Champions League and we have to fight at all costs."
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