A US think tank says the armed rebellion led by the head of the private military firm Wagner Group will likely substantially damage Russian President Vladimir Putin's government and his country's war effort in Ukraine.

The Institute for the Study of War says the rebellion "exposed the weakness of the Russian security forces and demonstrated Putin's inability to use his forces in a timely manner to repel an internal threat."

Media reports say Putin approached Belarusian President Alexander Lukashenko to resolve the rebellion. The Institute called this request "humiliating for Putin."

It says the imagery of Putin "requiring mediation from a foreign leader to resolve the rebellion will have a lasting impact."

The think tank says Wagner leader Yevgeny Prigozhin "likely gambled that his only avenue to retain Wagner Group as an independent force" was to rebel, after Russia's defense ministry ordered its fighters to sign contracts with the ministry by the end of June.

The institute also says Wagner forces may have shot down Russian forces' helicopters and aircraft, resulting in at least 13 military deaths. It described this as "one of the single deadliest days for the Russian air force of the war in Ukraine to date."