Yes, free will and determinism are compatible. It's known as Compatibilism, and most philosophers are actually Compatibilists. I often debate free will on reddit [1], so I'll reproduce the relevant argument here:
Firstly, to set the stage, understand that the "free will" debated in philosophy around questions of moral responsibility is not the same "free will" as typically used in science, eg. one such definition is where experimenters are free to set up their measurements independently of the system they're measuring. This is a common confusion.
What matters for the philosophical free will debate is whether there is a coherent definition of free will/choice which can make sense of our language of volition and moral reasoning, and which can serve to justify moral responsibility. Note also that "moral responsibility" does not necessarily entail "punishment" (which is a question of justice). This is another common confusion.
Finally, note that you make choices according to your nature, but to also have a choice in your nature would be logically circular.
So is there at least one definition of free will that can satisfy all of these criteria? Compatibilism is the most widely accepted approach among philosophers, and seems to match most people's intuitive moral reasoning [2]. For a rough example of what this might look like, consider a definition like "the moral responsibility of an agent capable of general learning is proportional to the amount of information it has learned, and a freely willed choice is one made based on internal reasons and not forced by other agents".
Note how this is perfectly compatible with determinism, how it makes sense of why we don't hold babies responsible but we hold adults responsible, it makes sense of how environmental [and] social factors that can impede learning moral lessons can diminish moral responsibility, it strictly defines what "free" means in the context of choice, and "moral responsibility" reduces to "moral feedback", ie. instructing what was done wrong. It's not perfect, but it should suffice as an introduction to Compatibilist-style reasoning.
This does not cover justice, which is what we must do in response to moral culpability, and which bring in further assumptions.
Can you elaborate more? I'm genuinely interested.