The US said that the city remained under government control.
Meanwhile, Zabihullah Mujahed, the Taliban spokesman, claimed in a series of tweets on Sunday that Zana Khan, Nawa, Jaghatu, Rasheedan and Khogyano districts surrounding Ghazni city have been overrun, and "multiple" security troops were killed and weapons seized. USA and North Atlantic Treaty Organisation forces remain in Afghanistan mainly in a supporting and training role in the 17-year-old war.
Baz Mohammad Himmat, head of the civilian hospital in Ghazni, said at least 16 people had been killed in the melee, including 14 soldiers and two residents.
The Taliban wants USA and North Atlantic Treaty Organisation forces to leave Afghanistan before participating in any intra-Afghan peace talks. The Taliban said they expect another round of talks.
The insurgents frequently exaggerate their battlefield gains and downplay losses incurred during clashes.
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Amid reports of resurgence of deadly clashes in the strategically important Ghazni city of Afghanistan, the country's army chief said on Sunday a full-scale clean-up operation is underway to eliminate the militants hiding in civilians' homes there.
The US presented it as a "failed attempt" to capture the city, and said that with close air support, Afghan forces were able to hold their ground.
Along a 3-mile stretch of the city, there were four Taliban checkpoints, each with a commandeered auto and a pair of motorcycles among four fighters, who appeared on their guard but unworried.
Separately from the Taliban, an Islamic State affiliate carried out dozens of deadly attacks in recent years, mainly targeting security forces and minority Shiites.
The insurgents have also so far ignored an offer by Ghani in February of unconditional peace negotiations.
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"The city was relatively quiet last evening and people were observed moving freely on the streets", Lt Col. Martin O'Donnell, spokesman for U.S. Forces-Afghanistan, said in an emailed statement. Washington has held preliminary talks with the insurgents in an attempt to find a negotiated end to Afghanistan's protracted war.
Instead, they appear intent on seeking a position of strength ahead of expected talks with the United States, which has been at war in Afghanistan for almost 17 years.
Pompeo also said the role of foreign forces in Afghanistan would be on the table.
The Taliban say they met with Alice Wells, the top USA diplomat for South Asia, in Qatar last month for preliminary talks.
Andrew Wilder, vice president of Asia programs at the U.S. Institute of Peace, said the attack by the insurgents was "a well-timed effort to demonstrate their military power to strengthen their negotiating position prior to another cease-fire and in the event of peace talks".
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We spoke at half-time about taking more responsibility for the second half, to do a little more. "After that, we will see". Despite signs of improvement in the second half, Emery's Premier League debut ended in defeat to the reigning champions.