A Chelsea statement did not mention the agreed transfer fee, but media reports say the deal is worth a club record 97.5 million pounds ($134.7 million).
The Red Lions beat Australia 3-2 in a penalty shootout after the final finished in a 1-1 draw.
Recent storms across parts of western Europe made rivers and reservoirs burst their banks, triggering flash floods.
Former England footballer Gary Lineker mocked the 31-year-old Lazio player.
Belgium were among the favourites for the tournament, having spent more than 1,000 days at the head of the FIFA rankings.
Italy set up a semifinal clash against Spain after first half goals from Nicolo Barella and Lorenzo Insigne led them to a 2-1 win.
The availability of both players has Belgium on tenterhooks ahead of the match in Munich on Friday after they were forced off the pitch injured in the last-16 clash with Portugal on Sunday.
Romelu Lukaku will likely be the main attacking threat when Belgium takes on Italy in Munich on Friday in the Euro 2020 quarterfinals. The Inter Milan forward has scored three goals in his team's four victories so far.
Switzerland meet Spain in the first quarterfinal on Friday (9.30 pm IST), while Belgium clash with Italy in a mouth-watering clash (Saturday, 12.30 am IST).
The winger scored a stunner against Portugal in Seville, ensuring Belgium a place in the European Championship quarterfinals.