The Afghan capital is on high alert ahead of an April 5 presidential election that Taliban insurgents have threatened to derail through a campaign of bombings and assassinations.
More than three hours after the initial explosions, sporadic gunfire could still be heard in the area, which had been cordoned off by police, a Reuters cameraman at the scene said.
Ghani, a former World Bank official, was not at home at the time of the attack.
The election is designed to mark the country's first democratic transfer of power. President Hamid Karzai is barred from running for another term after 12 years in power but is widely expected to retain his influence.
Initially police said militants had attacked Ghani's house itself but later Ghani's campaign aide said the assault started from an adjacent building housing a provincial election office.
"The attack was on an election office next to Dr Ashraf Ghani's house. His house was also hit by the attackers. He is not at home but his family is there," said the aide.
The interior ministry said there were two explosions followed by gunfire. There were no immediate reports of casualties.
Ghani, in Twitter post sent after the attack, said: "Just arrived at Gardez rally, a huge crowd of tens of thousands." Gardez is in a volatile area south of Kabul.
"A number of suicide bombers attacked an important election center at the heart of Kabul," the statement said.
"The attack started with a very heavy explosion and then a number of people entered with heavy and light weapons."
Separately on Tuesday, three suicide bombers entered a branch of Kabulbank, one of Afghanistan's biggest banks, in eastern Kunar province killing at least three security guards and wounding two bank workers, police said.
Taliban claims killing 20 people
The Taliban spokesperson, Zabihullah Mujahid, said a group of suicide attackers struck the election office in the Darul Aman area in the heart of Kabul.
Quoting eyewitnesses, the Afghan media monitored here said the suicide bombers entered an election commission office and correspondents in Kabul said that the attackers killed a guard at the gate allowing other gunmen to enter the building and start firing.
“Attacks started on an important center of the fake elections at 11 am. The attack was started with a big explosion at the gate,” the Taliban spokesperson said in a statement received by The Express Tribune.
Some reports said the attack occurred near the house the front-runner in the presidential elections, Dr Ashraf Ghani’s home. “Explosion and gunfire near Ashraf Ghani's house in Kabul were heard. But he is safe and in the eastern Paktia province for a campaign,” a journalist in Kabul told The Express Tribune.
Mujahid claimed the killing of 20 people and that the fighting is going on.
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