WASHINGTON — House Republicans on Thursday intensified their criticism of the Obama administration for its handling of the assault on the diplomatic compound in Benghazi, with Speaker John A. Boehner calling for the release of an e-mail that he said showed that State Department officials believed from the start that “Islamic terrorists” were linked to the attack but have declined to say so publicly.

Mr. Boehner’s demand, delivered the day after a dramatic hearing by a House committee, signaled that Republican leaders intend to continue portraying the attack as a major failure by the administration as well as a potentially effective issue against former Secretary of State Hillary Rodham Clinton if she runs for president in 2016.

“The State Department would not allow our committees to keep copies of this e-mail when it was reviewed,” Mr. Boehner said. “I would call on the president to order the State Department to release this e-mail so that the American people can see it.”

During Wednesday’s hearing, Representative Trey Gowdy, Republican of South Carolina, read an excerpt from the e-mail, quoting what he said was a reference to the local militant group that carried out the attack, Ansar al-Shariah, as having links to “Islamic terrorists.”