Sassoli, an Italian socialist and former journalist, died on Tuesday at a hospital in his home country, his office said. He was 65.
Polish President Andrzej Duda said on Twitter: "David Sassoli, President of the European Parliament, passed away during the night. I extend my deepest sympathies to his family, friends and associates. R.I.P."
Poland's Prime Minister Mateusz Morawiecki tweeted: "I was saddened to hear about the passing of David Sassoli, the head of the European Parliament. On behalf of the Polish government, I offer condolences to his family and friends. May he rest in peace."
Polish Foreign Minister Zbigniew Rau said in a tweet: "I am deeply saddened to hear about the loss of David Sassoli, President of the European Parliament - a great Italian and European."
Rau added: "My thoughts are with his family and all his countrymen."
Sassoli had been admitted to hospital last month because of serious complications related to his immune system, according to his associates.
Maltese lawmaker Roberta Metsola, from the conservative European People's Party (EPP), is widely expected to be a candidate to become the next president of the 705-seat European Parliament, the Reuters news agency reported.
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Source: PAP, Reuters