The clashes in the Nile Delta city of Mansura on Wednesday come amid widespread tension ahead of planned anti-Morsi rallies on Sunday to coincide with the first anniversary of his assuming power.
Two people were taken to the intensive care unit with live bullet wounds, said the Mansura International Hospital, where the patients are being treated.
A spokesman for the Muslim Brotherhood, Gehad al-Haddad, said one person was killed in the clashes.
Pro- and anti-Morsi gatherings are being held around the country, but confrontations have only erupted in Mansura, a security official said.
In Cairo's Tahrir Square, the epicenter of the 2011 revolt that toppled president Hosni Mubarak, hundreds of people gathered to watch the president's speech.
Some have announced they will begin a sit-in there.
Others have gathered outside the defense ministry, waving Egyptian flags and chanting against the president.