Doraiswami firmly put across to the US envoy that "it is your process and you have to sort it out", sources said.
New Delhi's assertion assumes significance in the context of the January 13 deadline for the indictment in New York of senior diplomat Khobragade on visa fraud charges.
The US said last month it is proceeding with the prosecution of Khobragade and has no intention of withdrawing the case of visa fraud against her.
India has been demanding the withdrawal of the case against her and an apology from the US for the treatment meted out to the 39-year-old diplomat, including a strip search and detention with criminals after her arrest on December 12.
The incident led to strong protests by the Indian government and widespread indignation in India. At the time of her arrest, Khobragade was the deputy consul general in New
York. She was subsequently transferred to India's Permanent Mission to the United Nations.
A 1999-batch Indian Foreign Service officer, Khobragade was arrested on charges of making false declarations in a visa application for her maid Sangeeta Richard. She was released on a $250,000 bond.
India retaliated by downgrading privileges of a certain category of US diplomats among other steps last month.
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