A tutor can be a helpful service for many students. A tutor service is often costly. Your child's 504 plan entitles him to reasonable accommodations. Section 504 does not specifically require schools to pay for a tutor.
Under Section 504, school districts will typically not voluntarily pay for private services of any type unless they are not capable of providing appropriate services using their own personnel. Unlike the IDEA, schools do not receive additional funding to pay for Section 504 services.
Read about the similarities and differences between the IDEA and Section 504.
Now, all that having been said, you may want to consider asking the school if there are tutoring programs already available in the district. Certainly, you can ask if the school is willing to pay for the private tutor, but understand that they are not required to do so if they can meet your child's needs another way.
Another consideration is, if your child is having significant learning problems despite the 504 plan, would a special education program be more appropriate for him? In special education programs, schools provide specially designed instruction for each child. While it may not necessarily be a private tutor, the school would be required to address your child's needs.
If you are willing to consider special education, you may want to consider requesting a meeting to discuss this possibility with the principal. Here is some information on how to make a referral for your child. I hope this is helpful to you. Please feel free to write if you have other questions.