The candidate for the 18 September elections, Khan Mohammad, running for a seat on a provincial council in Kandahar province, was captured along with a district chief and three policemen when they were travelling together in the south of the country on Friday, police said.
"We have killed them. We shot them," the Taliban spokesman, Abdul Latif Hakimi, told Reuters by telephone on Saturday from an undisclosed location.
There was no immediate official confirmation.
The Taliban, ousted by US and opposition forces in 2001, have condemned the parliamentary and provincial elections and claimed responsibility for attacks on several candidates.
Earlier, the Taliban spokesman said a Taliban council, or shura, would decide the fate of the kidnapped five.
More than 1000 other people, most of them fighters but including 49 US soldiers, have been killed in a surge of clashes, ambushes and bombings this year.
US and Afghan government forces have mounted a series of operations in the south and east in recent months aimed at clearing out fighters and ensuring security for the elections.
Afghan and US officials say the vote will not be disrupted.
Japanese identities confirmed
The two slain Japanese were
Meanwhile, an Afghan doctor on Saturday said two bodies found in Afghanistan are those of Japanese tourists who went missing along the Afghan-Pakistani border last month.
The bodies of a man and a woman were found in the desert near the main road between the southern city of Kandahar and the capital, Kabul.
The bodies were sent to Kabul on Friday to be examined.
"The dental records for the Japanese we received match," Eahsanullah Halemi, head of the hospital where the bodies were examined, told reporters.
"Both are the missing Japanese," he said.
The two were teachers - named by their school as technical arts teacher Jun Fukusho, 44, and female English teacher Shinobu Hasegawa, 30 - and were reported missing after they failed to return from a holiday in the region last month.
The pair had been shot and their bodies dumped near the Spin Boldak highway, which links the former Taliban stronghold of Kandahar with the Pakistani border town of Chaman, where the couple had crossed into Afghanistan on 8 August.