Soldiers attacked positions of the Iran-backed Shiite Huthi rebels from the east, west and north of the flashpoint southwestern city that has been under siege for more than a year, they said.
One military source said the troops made "a relative breakthrough" by driving the Huthi rebels from several positions around the city.
A pro-government activist, Abdel Majid al-Dhababi, confirmed that troops had made progress after months of trying to break the siege of Taez.
"But the road leading to Aden is still unsafe and this is preventing civilians from fleeing south," said Dhababi.
Yemen has been hit by unrest since the Huthi rebels and allied loyalists of former president Ali Abdullah Saleh overran the capital Sanaa in September 2014.
The violence increased after a Saudi-led Arab coalition launched a military campaign in March last year to help shore up the government of President Abedrabbo Mansour Hadi.
The government had fled to Saudi Arabia after the Huthis seized Sanaa and pushed south to second city Aden.
In March, loyalist forces backed by Saudi-led air strikes managed to break the rebel siege of Taez after fierce clashes but the insurgents later re-imposed a blockade on the city.
Tens of thousands of civilians are said to be caught in the fierce and protracted battle for Taez.
Saud-led aircraft meanwhile pounded rebels positions around Sanaa on Thursday, scoring hits on three military camps held by the insurgents, military sources said.
Violence on Thursday also rocked the southern Abyan province where an Al-Qaeda suicide bomber killed four Yemeni soldiers and wounded four others, a military official told AFP.
The bomber rammed his car into two military vehicles parked on a road linking the towns of Loder and Moudia, the military official said.
The attack came after troops recaptured both towns from Al-Qaeda in the Arabian Peninsula (AQAP), the official added.
He said the bombing was apparently in retaliation for the capture Sunday of provincial capital Zinjibar by the army backed by Saudi-led air strikes.
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