Rajih Badi told China's Xinhua News Agency the attack occurred near Basindawa's home.
"[An] unidentified gunman went out from a car and opened fire at the prime minister's convoy on a street close to the Political Security Agency in Sanaa," Badi said.
The prime minister's bodyguards returned fire but the would-be assassin fled, the adviser said.
Xinhua said an Interior Ministry spokesman said the ministry was investigating the attack. It did "not bear the hallmarks of al-Qaida ... but there is possibility that al-Qaida did it," the spokesman said.
Basindawa has been in office since November 2011 following the United Nations' brokered resignation of President Ali Abdullah Saleh after a period of unrest.
A six-month period of national dialogue for the country's political parties to draft a constitution and prepare for February parliamentary and presidential elections is set to conclude Sept. 18.
Aaron Carter is still in love with Hilary Duff
Kim Kardashian, Kanye West reportedly set wedding date