The rebels, known as Huthis or Ansarullah, had travelled to Amran to take part in a demonstration, but shooting erupted when they insisted on crossing a checkpoint to the northern entrance to the city with their weapons, an official said.
Eight rebels were among the dead, as well as two soldiers and two civilians, a local official and tribal sources said.
A senior security official in Sanaa said orders have been given to prevent Huthis from entering Amran, as the army dispatched reinforcements from the 310nd armoured brigade.
Last week, Huthis armed with assault rifles paraded through Amran along with vehicles fitted with rocket launchers, demanding the sacking of the "corrupt government".
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The Huthis have fought the central government in Sanaa for years, complaining of marginalisation under former president Ali Abdullah Saleh, who was ousted in 2012 following a year of protests.
Last month, President Abdrabuh Mansur Hadi and party leaders in Sanaa agreed to transform Yemen into a six-region federation as part of a political transition.
In early February, they seized areas of Amran province in fighting with local tribes that left more than 150 people dead.
Military sources said last week the rebel objective was to seize Amran city and, from there, lay siege to the capital.