Most of the fighting has been in a busy street where many Western companies have offices after Saudi security forces besieged suspected al-Qaida members holed up in the city's al-Hamra district.
"Three armed men were killed on Monday evening by security forces," a security source said, adding to two other fighters reported killed on Sunday.
Two policemen were also killed, he said.
But managing editor of al-Riyadh newspaper, Salim al-Ghamdi, told Aljazeera from Riyadh that three Saudi security men had been killed and 27 others injured in the clashes.
He said the number of fighters killed had yet to be confirmed.
"Saudi security forces are still surrounding the place where the terrorists are holed up in order to ensure the safety of local residents," al-Ghamdi said.
Three Saudi security men have
been killed in Dammam
"The terrorists have entered a nearby building and held nine residents hostage, including a handicapped father, his children and wife," al-Ghamdi said.
Earlier, the authorities described the besieged fighters as "members of the deviant group" - official terminology for al-Qaida fighters who have carried out a wave of shootings and bombings in the kingdom since May 2003, much of it targeting Westerners.
A security source on Monday said the authorities had no immediate plans to storm the fighters' redoubt but would instead lay siege to the buildings in an effort to capture them.
Explosions were heard in the area and ambulances rushed to the scene to evacuate an unknown number of casualties, witnesses said.
Saudi internet site al-Wifaq said one police officer had been killed and six wounded on Monday, although the police were unable to confirm the toll.
At least two police officers have
Two insurgents and an officer were killed in Sunday's firefight in Prince Muhammad bin Fahd Street, which houses the offices of many foreign companies operating in the kingdom's vast eastern oilfields, the Interior Ministry said.
Both of the slain men were sought by the authorities, the ministry added, without saying whether they were on a published list of most-wanted insurgents. One was captured but later died of his wounds, the security source added.
"Sporadic gunfire is continuing around the buildings where the members of the deviant group are holed up," a security source said.
"The security forces have reinforced their positions and are bringing in bulldozers and additional heavy equipment.
"We're in no rush to storm the buildings as we're hoping to capture them alive and obtain information."
Security forces have reinforced
their positions in Dammam
The US consulate in neighbouring Dharan said it would close for an unknown period of time because of the "standoff between Saudi security forces and an unknown number of terrorists in Dammam", according to a statement.
The violence in Dammam, a city 400km northeast of the capital, Riyadh, started at about 6pm on Sunday and continued into the night, sending scared residents running indoors.