Tribal sources told Reuters the rebels had gained control of Wadi Dhahr suburb in the fighting, the latest in an escalating conflict between the Sunni-dominated government and Houthi Shi'ites fighting for more territory and control in the north.
The conflict is further destabilizing a country struggling with a range of threats, including a separate secessionist movement in the south and an al Qaeda insurgency.
Fighting has also broken out in northern al-Jawf province, where 22 people were killed on Tuesday, following weeks of sometimes bloody anti-government protests by Houthi supporters in Sanaa.
Tribal sources told Reuters that at least 12 Houthi gunmen and gunmen from the Sunni Islah group were killed in Wednesday's clashes.
At least two medical sources said eight army soldiers died, after the Houthis stopped units heading to the suburbs to halt the rebels' advance. A Houthi leader said the rebels had destroyed two of the army's armored vehicles.
Residents in Sanaa reported hearing four explosions from the direction of the suburb early on Wednesday but the source of the explosions was not immediately clear.
The government had no immediate comment.
In the past weeks, the Houthis have capitalized on an unpopular decision by the government to reverse fuel subsidies to launch protests in the capital.
The Houthis said on Monday they would no longer take part in negotiations with the Yemeni government about a solution to their grievances because of what they termed "foreign intervention" in the course of the discussions.
Houthi protesters have been blocking the main road to Sanaa's airport and holding sit-ins at ministries to try to oust the government and restore subsidies cut by the state in July as part of economic reforms.
A government official told Reuters on Wednesday that the United Nation's special envoy to Yemen had gone to Saada province to meet the Houthi leader. REUTERS