Associates of Rakhat Aliyev, the former son-in-law of President Nursultan Nazarbayev who fell out with the veteran leader last year, started their own blog on Livejournal in June which often contains critical comments about the government.
"This is outrageous. They used to shut down papers and television channels, now they are shutting down the internet," a Livejournal blogger wrote in a posting.
A spokesman for Russian company SUP, which owns the website, said Livejournal itself was not blocking Kazakh users.
"The facts show there are no technical problems on our side," he said. "There has been no reaction yet from Kazakhtelecom or the Kazakh government."
Most of those who continue posting to Livejournal from Kazakhstan do so through smaller internet providers that still give them access to the blogging service or use websites that mask their location.
Kazakhtelecom said it had nothing to do with the issue.
"We will look into it, but we are not doing anything like that on purpose," Kazakhtelecom's IT Director Marat Abdildabekov told reporters.
Aliyev, sentenced by Kazakh courts in absentia to 40 years in prison for crimes including attempting a coup and kidnapping, lives in Austria where a local court ruled against handing him over to Kazakhstan. He says he is innocent.
Kazakh internet users have complained in the past about lack of access to websites the authorities do not like.
In 2006, a government-appointed internet regulator canceled the registration of website www.borat.kz used by British comedian Sacha Baron Cohen for his character Borat, a fictional Kazakh journalist deemed offensive by some local officials.
(Reporting by Masha Gordeyeva; additional reporting by Alexander Gelogayev in Moscow; writing by Olzhas Auyezov; editing by Tim Pearce)