"Touch base" vs "Touch bases"?

"Touch base" is the significantly more common usage and is listed in phrase dictionaries. The general impression is that it comes from the sport of baseball:

In baseball, a player who is touching a base is not in danger of being put out. Another explanation is that a player briefly touches each of the bases when he runs around after hitting a home run; therefore "touching base" is briefly checking in (this is more similar to the meaning in the above example).

The above quote is unsourced from Wikipedia, so it isn't terribly authoritative.

I would use "touch base" as it is the clear winner in terms of usage and definition. "Touch bases" isn't going to be misunderstood but it appears to be the derivation.

Touch base is by far more common than touch bases:

It comes from baseball from when you touch bases as you proceed around the baseball diamond.

It means that you got or will get in contact with somebody:

Alan: Did you talk to Mark yesterday?

David: No, but we'll touch base later in the week.

Touch base is more appropriate in the sense that base is singular in context. You are not touching each other's respective bases together, you are both touching one singular and common base as a point of foundation or contact. Similarly, you wouldn't say let's make contacts tomorrow if you were only speaking to one other person.

It may be possible that touch bases is a corruption of basis, which is a synonym for base, but for what it's worth, I have never heard anyone say touch bases.