It has sold more than 680,000 copies in its first three days alone, beating Fifty Shades of Grey which sold 664,478 in a single week in 2012.
At its current rate, it is on track to become the second biggest single-week sales for a book since records began.
That title is currently held by Harry Potter and the Deathly Hallows, according to The Bookseller.
The final novel in the Potter series sold 1.8 million copies - as well as 780,000 copies of the version aimed at adults - in its launch week in July 2007.
Harry Potter and the Cursed Child has also become the fastest-selling script book, said publishers Little, Brown.
Meanwhile, the script has also sold more than two million print copies in North America in its first two days, according to publishers Scholastic.
Reviewers have complained the script is an "incomplete experience" as the story "demands to be seen", but theatre critics awarded the play five-star reviews.
The stage production in London's West End is currently sold out, but a new batch of 250,000 tickets is being released on Thursday.
Written by Jack Thorne and approved by Potter author JK Rowling, the production focuses on Harry and Draco Malfoy's sons - Albus and Scorpius.
Kate Skipper, buying director at Waterstones, said she was certain the script would be the chain's "biggest book of the year".
She said: "Our sales for Harry Potter and the Cursed Child script book have been phenomenal. We saw our biggest first day figures since Harry Potter and the Deathly Hallows was released in 2007 and after just two days' sales Harry Potter and the Cursed Child is already our biggest-selling hardback since Dan Brown's The Lost Symbol in 2009.
"By the end of this first week, we expect to exceed The Lost Symbol sales and to match the lifetime sales of our bestselling script book ever, An Inspector Calls by JB Priestley."
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The Harry Potter and the Cursed Child script has become the fastest-selling book in the UK this decade.