Barcelona fans have no collective feelings towards Inter. Since there are fans from everywhere now, it's just personal or regional biases that take place. So a Barcelona fan from the USA will have more feelings towards PL teams than Inter because his surroundings are focused on those teams. But speaking of Barcelona proper fans, there are some 'rules' let's say.
Barcelona dislike Real Madrid, Espanyol and Athletic Bilbao. That does not change, ever.
From foreign teams their main rival was actually Milan since both were competing for the Champions League in its early years and Milan gave them a beating despite Barcelona being considered the heavy favorites (4-0 final in Athens). And this carried on in the mid 2000s as well.
Which is why most "old school" (ffs, I'm talking about the 90s here haha) Barcelona fans liked Inter and which is also why I felt a stronger connection with them than with Real Madrid when I "picked" my Spanish team (I stopped 'supporting' Barcelona in Spain in 2009 but until then I tried not to miss a game). But the newer generation is probably not aware of a heated situation with Milan but have more animosity towards us than before as we "stole" Ronaldo and we've beaten them from the Bernabeu final and all that. But generally, they don't really care about us just as we don't really care about them. Mutual respect as a great European club, that's about it.
Chelsea fans hate Barcelona but I doubt Barcelona fans even care that Chelsea exists. The same applies with Celtic iirc.
From my memory I remember Barcelonistas disliking Man Utd the most from England and Bayern from Germany. Juventus is a weird one as they were their main 'hate' team from Italy but when Berlusconi's Milan came along the new generation started liking Juventus more and then the Old Bitch had a rivalry going with Real Madrid which just helped to that. So some Barcelona fans between the ages of 30-40 have a strong liking for Juventus which means by default they dislike us. With 2010, that just became even more collective for some but I wouldn't say they think about us all the time.
I'd compare it to us vs Liverpool maybe.