Loud explosions from the attack could be heard and debris could be seen flying, during the 90-minute firefight. Personnel from the CIA station near the palace were engaged in the fight, the BBC also reported.
The presidential palace is reportedly one of the most secure areas of Kabul.
The attack is likely to further strain efforts to launch formal peace talks between the U.S. and Taliban, first announced last week.
The assault came as a U.S. envoy met with representatives from President Hamid Karzai’s government to smooth relations.
The White House announced last week that it would begin formal negotiations with Taliban officials in Qatar. But that effort faced pushback from Karzai, who said it represented an unwarranted shift in U.S. policy and retaliated by stalling talks with Washington on a post-war security deal.
Secretary of State John Kerry on Saturday called on the Taliban to help get the talks “back on track” after further delays.
The U.S.-led NATO coalition is pulling all combat forces from Afghanistan in 2014, leaving only support troops. Earlier this month, allied forces formally handed over security responsibility for the country to Afghan troops.