Al-Nusra Front, which has previously executed one captive Lebanese soldier, issued its threat after troops cleared militants from Tripoli's historic central market district yesterday in an operation that left one civilian and one militant dead.
Many of the militants managed to withdraw to the Bab al-Tebbaneh district of the city, a Sunni Islamist stronghold, where troops were engaged in heavy fighting today, an AFP correspondent reported.
The army's offensive against the militants - who are suspected of having links to Al-Nusra - sparked attacks on troops across the Tripoli region that left six dead.
"We call on it to lift its siege and accept a peaceful solution, or else we will be forced in the coming hours to bring closure to the issue of the soldiers we are holding hostage, given that they are prisoners of war."
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The Al-Qaeda affiliate initially threatened to start executing its prisoners from 0130 IST but then issued a second statement extending the deadline to 0530 IST.
"At that time the execution can still be postponed or cancelled if the army agrees" to our demands, the later statement said.
IS has since executed two of its captives.
Al-Nusra has previously demanded that in return for the release of its prisoners, Shiite militant group Hezbollah end its intervention in Syria on the side of President Bashar al-Assad's regime and that Lebanon free jailed Islamists.
The Lebanese government has so far rejected the terms.
Islamist gunmen in Tripoli have carried out repeated attacks against the army, accusing it of cooperating with Hezbollah.