Iranian newspapers reportedly carried a letter to Mr Mousavi from Iran's police chief, Ismail Ahmadi Moghaddam, in which he warned that officers would "decisively confront" any further unrest.
Witness accounts suggested police used live rounds, batons, tear gas and water cannon to break up demonstrations which went on late into Saturday night.
A BBC reporter said he saw one man shot and others injured amid fighting, with streets littered with burning debris.
There were reports of deaths from both protesters and official sources.
A state TV report said "rioters set on fire the Lolagar mosque... in which unfortunately a number of our citizens were killed", but there has been no other word on the incident.
The protests went ahead despite a direct demand from the country's Supreme Leader, Ayatollah Ali Khamenei, to end them and accept President Mahmoud Ahmadinejad's victory in 12 June presidential elections.
In a statement on his website, Mr Mousavi accused authorities of cheating the Iranian people out of a rightful election result, reports said.