Several analysts looking at the situation in Russia at the time of the coup attempt saw the event as an embarrassing blow to Putin's authority. Many also predicted that Prigozhin's days were likely numbered.
"I think a lot of the criticism that Prigozhin was making … was not that they were against the invasion of Ukraine, [but] they were against the failure to win in Ukraine," Bolton told CNBC.
"Putin's hand I think, is strengthened now. And I think we're at a very critical period in the battlefield in Ukraine because of the stalled spring offensive by the Ukrainians."
The former national security official and American ambassador to the U.N. also expressed concern over the slow progress of the long-anticipated Ukrainian counteroffensive, which Kyiv and its allies had hoped would decisively turn the tide against Moscow.
"I'm very worried, frankly, from that point of view, that now, especially, the Kremlin will take the advantage of September, maybe early October, to try and move diplomatically to achieve what they've not achieved on the battlefield," Bolton said. "So I think we're in for a perilous time ahead here."
Read more of CNBC's politics coverage:
Putin expressed his condolences following Prigozhin's demise in a televised address Thursday.
"With regard to this plane crash, first of all, I want to express my sincere condolences to the families of all the victims. It's always a tragedy," Putin said.
"I've known Prigozhin for a long time, since the early '90s. ... He was a talented man, a talented businessman. He worked not only in our country — and with result — but also abroad," he added, referencing Prigozhin's contracts in Africa.
The Russian president added that an investigation into the explosion that brought down the private jet was already underway.
— CNBC's Karen Gilchrist contributed to this report.
