Reports say the script is from episode four of the seventh series, which is due to air later this week.
Some programmes were also accessed in the cyber attack, although a spokesman for the company would not specify which shows the hackers had stolen.
In an email to staff, HBO’s chief executive Richard Pepler said hackers had taken “proprietary information”.
The company said it was working with police and cyber security experts after the incident.
Entertainment Weekly magazine published an email sent to reporters on Sunday purporting to be from the hackers, promising an interview to anyone who “spread the word” about the hack.
When Sony was hacked in 2014, films and emails sent between employees were leaked, and in April this year, a hacker claimed they had leaked episodes of Orange Is The New Black before Netflix streamed them.
The anonymous email sent to Entertainment Weekly called the hack “the greatest leak of (the) cyber space era”.
It said: “Hi to all mankind. You are lucky to be the first pioneers to witness and download the leak. Enjoy it & spread the words. Whoever spreads well, we will have an interview with him.
“HBO is falling.”