No, you cannot apply for a Luxembourg visa without having a valid job offer. One of the requirements to obtain a Luxembourg work visa is to have an employment offer from a verified employer in Luxembourg. The employer must undergo a Labour Market Test to prove there is no local talent or candidate to fill the vacant position. The employer then begins the employment process at the ADEM and grants a job offer to the employee. Once the employee receives a job offer from the Luxembourg employer, they can apply for a work visa and migrate to Luxembourg.