Is it correct to say express purpose or expressed purpose?

The correct phrase is "express purpose." "Express" is an adjective that means clear, specific, or direct, and in this context, it is used to describe the purpose as being clearly stated or defined. "Expressed" is the past tense or past participle form of the verb "express," and it would not be used in this phrase as it does not convey the same meaning as "express" in this context.