At least one processor core has debug and non-debug modes of operation. Debug control circuitry controls operation of the at least one processor core when in the debug mode. On power up of a given processor core, the core checks a debug status value stored in a debug status storage element. When the debug status value has a first value, a debug connect sequence of messages is exchanged with the debug control circuitry over a debug interface to determine whether the given processor core should operate in the debug mode or the non-debug mode, and the debug status value is set to a second value when it is determined that the given processor core should operate in the non-debug mode. When the debug status value has the second value, the given processor core omits initiating the debug connect sequence and determines that it should operate in the non-debug mode.