The Expressen newspaper reported on Wednesday that a group called "Afghanistan's Islamic Movement Fidai Mohaz" released the statement on its website and Facebook page. Its spokesperson Qari Hamza wrote that the group had killed Horner, 51, on Tuesday when the reporter was shot from behind while doing interviews on the street in central Kabul.
"He was not a journalist, he is a spy for MI6," the text on the website claimed.
Christian Nilsson, chargé d'affaires at Sweden's embassy in Kabul, said the embassy would not comment on rumours.
"There is a lot of speculation right now with a lot of rumours," he told Expressen.
The Taliban on Tuesday said they were not behind the murder.
READ ALSO: Swedish radio reporter gunned down in Kabul
Reports came through early on Tuesday that a foreign national had been killed in the Afghan capital near a restaurant where 21 people, including 13 foreigners, were murdered in January.
Sveriges Radio (SR) later confirmed that the man was their South Asia correspondent Nils Horner. SR added later that the 51-year-old Swede was shot in the back of the head while conducting street interviews.
"This is one of the worst days in the history of Sveriges Radio," said SR's CEO Cilla Benköö. "Nils was one of our absolute best and most experienced correspondents."
Urban Hamid, an experienced Swedish war correspondent, said it was easy to downplay the risks of working in Afghanistan when spending time in the capital.
"When you are out and about in Kabul you usually feel quite safe," Hamid told The Local. "But maybe if feels safer than it actually is, because there are a lot of soldiers, police officers and checkpoints everywhere."
"Foreign reporters and aid workers are easy targets as they move about without guards," he added. "The problem now is that people have come to realize you can kill a journalist without any repercussions."