Is it a thing? It’s not really a thing. But it’s kind of a thing. Nothing has been signed and if it was, no one really knows what it would be. A small investment? A huge one? A takeover? We don’t know. We just don’t know.
We DO know that some talks have taken place about an arrangement between the companies and that idea of those two powerful brands teaming up is pretty interesting. Apple has more money than all the rest of the money in the world combined (that’s a rough estimate) and Twitter, while culturally ubiquitous, doesn’t have all that much money, actually. And lately, the old Fail Whale has been showing up again when the servers are insufficient to handle the demand.

The Times says that the talks are paused but may resume.
From The Times:

Though an investment in Twitter would not be a big financial move for Apple by any stretch — it has $117 billion in liquid investments, and it quietly agreed to buy a mobile security company for $356 million on Friday — it would be one of Mr. Cook’s most important strategic decisions as chief executive. And it would be an uncommon arrangement for Apple, which tends to buy small start-ups that are then absorbed into the company.
But such a deal would give Apple more access to Twitter’s deep understanding of the social Web, and pave the way for closer Twitter integration into Apple’s products.