"It poses a challenge to Putin. For all his macho persona, Putin's a guy who doesn't like making tough decisions, he tends to put them off as long as possible, often too long," Galeotti said.

"Is he going to essentially deal with Prigozhin ruthlessly as essentially a traitor? Which let's face it that is what this was. This was treason. Or does he look for some kind of fudge or pretend it's not an issue," Galeotti said. He said that if Putin chooses to ignore the issue, "this will reinforce the views of quite a few people within the elite who are thinking that essentially he's lost his way and is no longer the Putin as once was."

Galeotti partly blamed the political system that Putin has created for the feud, which he said is a system of "divide and rule" that has resulted in "a whole variety of sharp-toothed political entrepreneurs with overlapping responsibilities competing against each other, trying to rip each other down." He said that system has worked well for Putin politically, but "really doesn't work well when it's transposed to the battlefield and you have all these multiple different sorts of private armies and different forces."

Kateryna Stepanenko, an expert on the Russian military at the Institute for the Study of War, told Insider's Ryan Pickrell that Putin having Prigozhin killed would '"undermine his appeal to the nationalists, who are the only people that are so inherently invested in his ideology and his belief in this war."