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The Federal Bureau of Prisons inmate records on Sunday showed him held at the Metropolitan Detention Centre in Brooklyn.
Modi has denied any wrongdoing and his lawyer declined to comment.
The source added that Mallya's extradition is before India's foreign secretary as an important matter seeking immediate attention.
The leave to appeal to the Supreme Court is on a point of law of general public importance, which according to experts is a very high threshold that is not often met.
A CBI source related to the development said that after Mallya's extradition, they will take his custody first as they were the first agency to file a case against him.
The 2008 Mumbai attack was one of the most horrific terrorist attacks in India in which 166 people were killed and over 300 injured as 10 heavily-armed terrorists from Pakistan created mayhem in the country's financial capital.
Britain's court rejected Indian liquor and aviation tycoon Vijay Mallya's plea against extradition .
Modi's barrister Hugo Keith informed the court of an extortion attack when two inmates entered his cell and kicked him to the floor and punched him in the face.
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