Is it permissible for a manager to take a percentage from engineering consulting and contracting offices he deals with, in exchange for providing the owner with lower prices of the same quality, whether this percentage is from the owner's projects or from projects of other individuals, knowing that this percentage does not exceed the original agreed-upon price?

It is permissible to take a percentage or commission from offices if you bring them a project from outside your company, and this is considered brokerage. However, if your work is a duty and you receive a known salary from your company for it, then it is not permissible to take a percentage unless the company grants you permission to do so. Your duty is to exert effort for the benefit of your company in exchange for your salary.