I've been living about six months abroad on a student visa.
Currently, I am receiving many offers to work here, but I'm always blocked because my visa limitations.
How could I approach the headhunters with the situation with regarding my visa?
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Many employers can sponsor work visas, so just ask the headhunters if this is an option. However, you'll need to finish up your studies first.
Depending on the country you're in, it may be possible to legally work while you study, but this typically comes with tight limits on how many hours per week you can work and what kind of work you can do, meaning this is usually only an option for part-time retail jobs or similar.