I can confirm that you are allowed to work for a foreign company as an Italian resident permit holder. I assume you are a neo-domiciled individual, so you are entitled to opt for your non-Italian sourced income to be taxed in Italy through the application of a flat substitutive tax, at a fixed amount of 100,000 euros.
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Do I need to pay tax for my foreign-sourced income as a resident in Italy?
I obtained a resident permit in Italy through an investment in a startup company and have been living in the country for a while. Now I have been offered a remote position by a company in the U.S. Am I allowed to work for a foreign company as a resident permit holder? How is my foreign-sourced income taxed?