Muntazer Al Zaidi gained instant international fame when he threw his shoes at Bush during the US president's farewell visit to Iraq on December 14, an action considered a grave insult in the Arab world.
"At the beginning of the month his family contacted me via the ICRC (International Committee of the Red Cross) and I shall write this week to the federal department (ministry) of foreign affairs to encourage Switzerland to grant him political asylum," lawyer Mauro Poggia told the paper.
Switzerland could give him asylum "without taking a position for or against the American intervention in Iraq," he said.
Zaidi, a 29-year-old journalist for the Al-Baghdadia television channel, had been due to appear before Iraq's Central Criminal Court on December 31 on charges of "aggression against a foreign head of state during an official visit" and he faced up to 15 years in jail if convicted.
But the court decided to postpone the trial pending an appeal to a higher tribunal.
Even if many Iraqis supported his action, Zaidi was "at the mercy of all manner of extremists," the lawyer said, adding: "He can no longer work as a journalist without suffering terrible pressure... his life can become hell in his country."
Once settled in Geneva, the bachelor without children could "very well work as a journalist at the United Nations" which has its European headquarters here, Poggia said.
After throwing his shoes at Bush, Zaidi also insulted the American president, shouting: "It is the farewell kiss, you dog."
Zaidi missed Bush, who ducked the throws, but Iraqi and US security officials grappled with him and hustled him off while the US president tried to joke his way out of the incident.
The journalist's action was hailed across the Arab world as an ideal parting gift to a president widely unpopular in the region.