Three Saudi men on a list of 23 people wanted by the authorities over security offences have turned themselves in, the interior ministry said on Monday.
The report, carried by state news agency SPA, came as Saudi security forces pushed ahead with an operation in the eastern part of the kingdom to try to flush out armed men, including those on the list announced in January 2012.
The interior ministry identified the three as Mohammad Esa Al Lubbad, Ramzi Mohammad Jamal and Ali Hassan Al Zayed, and said their "initiative (to surrender voluntarily) will be taken into consideration".
Many of those on the list have been either killed or captured in recent years.
The Saudi Gazette newspaper said that only three of those on the original list remained at large, while eight have surrendered.
The rest were killed during clashes with the security forces, it said.
Saudi security forces have been trying for more than two months to defeat gunmen behind attacks on police in Awamiya, a town of around 30,000 in the eastern region that has been the centre of protests against the government.
Fighting has intensified over the past two weeks, when elite forces entered the town.
In May the authorities began a campaign to tear down the old quarter to prevent gunmen using the narrow streets to evade capture.
Residents estimate that up to 20,000 people have fled to towns and villages nearby.
Up to 12 people have been killed in the past week: three policemen and nine civilians, residents say.
The area, in oil-producing Qatif province, has seen unrest and occasional armed attacks on security forces since 2011.