The telecoms tycoon first announced the initiative on Twitter.
"Greece or Italy sell me an island, I'll call its independence and host the migrants and provide jobs for them building their new country," he wrote.
More than 2,300 people have died at sea trying to reach Europe since January, many of them Syrians who fled their country's four-and-a-half year conflict.
Sawiris said in a television interview that he would approach the governments of Greece and Italy about his plan.
Asked by AFP whether he believed it could work, he said: "Of course it's feasible."
"You have dozens of islands which are deserted and could accommodate hundreds of thousands of refugees."
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Sawiris said an island off Greece or Italy could cost between $10 million and $100 million, but added the "main thing is investment in infrastructure".
There would be "temporary shelters to house the people, then you start employing the people to build housing, schools, universities, hospitals.
"And if things improve, whoever wants to go back (to their homeland) goes back," said Sawiris, whose family developed the popular El Gouna resort on Egypt's Red Sea coast.
He conceded such a plan could face challenges, including the likely difficulty of persuading Greece or Italy to sell an island, and figuring out jurisdiction and customs regulations.
But those who took shelter would be treated as "human beings," he said. "The way they are being treated now, they are being treated like cattle."
Sawiris is the chief executive of Orascom TMT, which operates mobile telephone networks in a number of Middle Eastern and African countries plus Korea as well as underwater communications networks.
He also owns an Egyptian television channel.