The IOC suspended the IBA in 2019 over finance and governance issues and did not involve it in running boxing events at the subsequent Tokyo Olympics either.
The strained relation between the sports bodies further soured after Russia's invasion of Ukraine last year.
The amateur boxing's governing body defied IOC guidance and lifted a ban on the Russian and Belarusian boxers competing under their flags last October.
Kremlev said IBA under him had put corruption and bankruptcy behind itself to emerge as an efficient organisation.
"The IBA has fulfilled all the criteria 100% concerning all the questions that were raised in relation to judging and system reforms," the Russian told Reuters on the sidelines of the ongoing women's world championship in New Delhi.
"We have no debt, we have a good financial system in place. We have active and former boxers in our board of directors.
"I think that we are one of the 10 most efficient and successful international associations in the world."
The IOC plans to keep IBA out of the qualification process for next year's Olympics in Paris too, while boxing is not on the initial programme for the 2028 Los Angeles Games.
Kremlev said any move to drop boxing from Olympics would be the "most criminal decision in the history of Olympic movement" which would face a backlash.
"We are not going to let anyone to take boxing out of the Olympic Games," he said.
"Those who want boxing to be excluded from Olympic Games will be considered criminals.
"Millions of boxers are ready to come to Lausanne (Switzerland) and ask for explanations from IOC, because we are one strong family, a united family.
"If they're going to make such a decision, I think IOC will have to be reformed."
Kremlev called the IOC-IBA standoff a "misunderstanding", which could be resolved in a day.
"I really hope that everybody will come to their senses and instead of exchanging mails, we could just sit down at the negotiation table and find a solution," he said.
"I think the solution could be found in one day. I think that's possible."
But for that, said the Russian, the IOC would have to treat the IBA as "equal partner".
"We should not be told what to do, we should not be dictated what to do. We need to be dealt with equally, as an equal partner."
