Yes, both are okay. The one with "were" is mildly odd, but I'd call that a style problem. The context kind of encourages use of "were". In "If you {were/*are} rich you could afford this" and "If you {are/*were} rich you can afford this", the distribution is more strict, and corresponds to counterfactuality in the case of "were". As an isolated answer to a question like "Is this affordable?", selection of were vs. are in the response ("Sure, if you {are/were} rich") similarly correlates with the implication that we don't know if you are rich (are) vs. we know that you are not rich (were). By selecting "were" in your pair, you are suggesting that the addressee is not free, which is a way of defeating the inference that you've just imposed an obligation on the addressee (since the sentence is plainly a request for a meeting). Perhaps my judgement of oddness stems from my dislike of contorted ways of making requests.

I agree that the first sentence comes over as not quite right.

I think we can show the problem with two different sentences:

We could go to the cinema if you were free.

The speaker acknowledges that the person addressed is in fact not free.

We could go to the cinema if you are free.

Here the speaker is uncertain if the person addressed is free.

Actually, I should clarify one thing: if these are questions as written in the original post, then "were" is very strange, assuming that's like a question tagged on the end-- "... If you were free?" But if it's a statement, "I thought X, if Y were the case" then that's OK as an alternative.

I think it is a case that if you saw, and more likely if you heard: "I thought maybe we could have a little chat now if you were free" whether as question or statement, it would pass unnoticed. It is only when asked to consider it, and especially if asked to compare it with: "I thought maybe we could have a little chat now if you are free" that you feel that, even if the former is not wrong, the latter is better.

I would not rely much on intonation? If you know what I mean?

Based on what you have said here, I assume that the sentence "I thought maybe we could have a little chat now if you are free?" sounds better (especially when we are addressing our friend etc.) and is considered correct grammar? On the other hand, the past form "were" could be used if we wanted to sound sort of less direct and more polite?

I just wasn't sure if we could combine present and past forms together in a "if" sentence, but apparently we can. There is even the following sentence in my notebook (the lesson was about mixed conditionals and the sentence was given by my teacher as an example of 1 + 2 conditional, where we mix the hypotheticity):
I'm giving you the money if you wanted to buy something.

OK I'll try to think this way. But just to be sure, when we are talking about grammatical correctness, can we say in this particular case than one form is *more* correct than the other, or not really?

I just wasn't sure if we could combine present and past forms together in a "if" sentence, but apparently we can.

The point is not so much that the sentence with "were" can be classed as ungrammatical, but that (especially when asked to consider it) it seems to have lost its way with the two halves being out of sync. Panini hits the nail on the head when he says: 'By selecting "were" in your pair, you are suggesting that the addressee is not free, which is a way of defeating the inference that you've just imposed an obligation on the addressee (since the sentence is plainly a request for a meeting).'