Is it permissible for a government employee to accept an offer from a private company—which includes renting accommodation for him and paying an additional salary—in exchange for his supervision of its project, knowing that this company is under his supervision and that the government does not reimburse him for these expenses?

If the head of the construction team works for the company that will supervise it, and your government employer is unaware of this agreement, then it is not permissible for you to do so. This is because receiving an additional salary and housing might affect your supervision and cause you to overlook their mistakes. Prohibited gifts include anything given to workers and those in positions of authority that influences their work.

However, if this is done with the agreement and knowledge of your employer, then there is no objection to it. An employee must fear Allah in their work and avoid bribes and favoritism.