AIIMS reacted to the allegation, saying that it was a matter of concern and it was looking into it.
They are also learnt to have informed the Crime Branch of Delhi Police about the issue.
Rai, in a series of tweets, posted images of question papers of the entrance examination which was held across the country on May 28.
He said that he had received the question paper snapshots from a source who claimed these were leaked from M C Saxena College in Lucknow when the online test was on.
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The AIIMS online entrance test was held in two shifts, he told PTI, adding that the snapshots clearly prove that the question papers were leaked while the examination was going on.
"I urge the prime minister and the AIIMS director to ask the CBI to probe the matter," he said.
Rai also alleged that AIIMS MBBS seats were being "sold".
In his Facebook page, he has urged Prime Minister Narendra Modi to ensure justice for meritorious and hardworking students who suffer because of such "mafias".