When you're having lunch and you see someone he can say "enjoy your meal", "bon appétit" or "enjoy". I can answer him by saying "thank you", for instance. But for example in Spanish we usually say a sentence which means something like "you can join me/us, if you want" (inviting him). Is there any typical sentence in English with a similar meaning?
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No, I don't think English has a comparable common phrase.
However you could always just ask him if he's interested in joining you.
Since there is no set phrase/cultural equivalent in English for Bon Appetit (you can say "Enjoy your meal", and it would be understood, but it is not what people say' when starting a meal and is pretty uncommon), there is even less likely an accepted reply.
If I were to describe a classic British meal, the appropriate response to something so unexpected would be stony silence and reflection upon ones past sins as though one were climbing the stairs to the gallows. Any more substantive response would get nervous glances and an inner reminder to avoid playing darts with them.
But one could have a smoother, close to literal translation that could work (e.g. something like "Please join us"). It just wouldn't be idiomatic.