Arthur resigned and the entire selection panel were sacked last week following the drawn series with England, while Kirsten's current two-year contract comes up for extension in March.
Media reports have suggested that Kirsten will be offered the job at the end of South Africa's current Test and one-day tour of India.
"He will continue his stint with the Indian team," Board of Control for Cricket in India (BCCI) president Shashank Manohar told DNA newspaper on Wednesday. "We'll renew his contract."
The extension would take Kirsten's tenure up until next year's World Cup which India will co-host with Sri Lanka and Bangladesh, the paper said.
Former fast bowler Corrie van Zyl has replaced Arthur on an interim basis for the current Test and one-day tour of India.
"I have not been contacted by (Cricket South Africa)," Kirsten told the paper.
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