The 31-year-old sustained the injury less than ten minutes into Saturday's defeat at Lancashire rivals Preston North End.
A scan on Monday confirmed that the former Trinidad & Tobago international had suffered a tear to his hamstring and is likely to be out of Stan Ternent's line-up for up to four weeks.
On the injury setback to Cox, Ternent told The Clarets` official website: "The scan showed he has a torn hamstring and he will be out for three weeks to a month.
"The centre of the defence is a problem area for us at the moment.
"We have Gordon and Coxy out injured and that leaves us with Steve Davis, Arthur Gnohere and Mark McGregor, so picking that part of the team hasn't been difficult."
