The 18-year-old England Under 21 international goalkeeper, who joined the Robins on an initial one month loan deal last month, will now remain with Mark Yates's side until 11th December.
"We wanted to keep Jack as long as we could and Birmingham have been brilliant," Yates told the official Cheltenham website.
"They appreciate he's getting games and the lad is really enjoying it.
"He's getting a taste of league football and the agreement suits both ourselves and Birmingham."