Kazakhstan Today said both suspensions were due to violations in Kazakh law and applied for three months.
Police and the general prosecutor's office were not available for comment.
"This is complete lawlessness," Karavan chief editor Alexander Shukhov told Kazakhstan Today.
"We are dealing with the fact of police despotism which is on the brink of a police coup."
Aliyev, Kazakhstan's ambassador to Vienna, has not been available for comment. The Kazakh embassy in Vienna has declined to comment on the matter.
On Wednesday police were ordered to investigate Aliyev on suspicion of kidnapping two senior officials of a bank he owned.
Analysts said the move was part of Nazarbayev's bid to tighten control in the oil-rich Central Asian state.
Earlier in the week he signed constitutional amendments that paved the way for him to stand for election as many times as he wanted and theoretically become president for life.
Aliyev has previously suggested that Kazakhstan could become a monarchy and has been seen as a rival to Nazarbayev, where clan divisions often dominate politics.
His wife, Nazarbayeva, is a powerful figure in her own right, controlling the state's biggest media holding and has also been seen as potential successor to her father.
Separately on Thursday, police detained Sergei Duvanov, an independent journalist, for protesting on the main square of the financial capital Almaty against this week's constitutional amendments.
Duvanov has frequently been arrested by authorities and convicted on a rape charge that he claims was trumped up by authorities.