CAIRO, June 30 (Reuters) - Egyptian President Mohamed Mursi knows he has made mistakes and is working to fix them, a spokesman for the Egyptian leader said on Sunday after a day of mass protests demanding that the head of state quit.

Presidential spokesman Omar Amer said Mursi was serious in his repeated calls for national dialogue.

"(Mursi) announced to all of Egypt's people he made mistakes and that he is in the process of fixing these mistakes," Amer told a late-night news conference after millions took to the streets to demand Mursi leave power.

He added that Mursi had "extended his hand" for dialogue and wanted to listen to everyone, repeating the president's previous calls for national dialogue which the opposition has rebuffed as not serious.

"I want to confirm one truth, if there is a total lack of response to this initiative, no listening to it, no interest in it from any side, what do you think the presidency can do?" the president's spokesman said.

"The presidency is now waiting for a reaction, no matter how small, so it can build on it."

(Writing by Tom Perry; Editing by Paul Taylor)