Called "Dirty Corner" but dubbed the "queen's vagina", the 60-metre (200-foot) long, 10-metre high funnel-like sculpture has been repeatedly vandalised since it was unveiled in the palace's gardens in June.
When anti-Semitic graffiti was daubed on it this month, Kapoor said he wanted the "abominable words" to be left.
But the 61-year-old British-Indian superstar artist has had a change of mind.
The authorities who run the palace outside Paris said work will begin in the next few days "to cover up the damage, under the supervision of the artist".
Kapoor's work is not the first to be defaced recently in France.
In October 2014, vandals in Paris's chic Place Vendome deflated a massive sculpture by American artist Paul McCarthy that was shaped like a sex toy.