Is it 'Close to the chest' or 'Close to the vest'?

Either one is fine. Close to the vest has a more British feel to me, but I've heard both in the U.S.

EDIT

OK, since this apparently bothers someone, I did some research and it appears that the "vest" usage is more American. Note that my original statement of "British feel" was admittedly idiosyncratic.

Note also that either one is still fine, despite any individual's peevish disapproval.

Here are some NGram searches and their results:

British English:

American English:

Combined British/American:

Apparently the "vest" version came into British English in the late 1950s, but didn't gain widespread acceptance until the 1990s. Currently it looks poised to gain equivalence with the "chest" version, although such things are hard to predict.

Disclaimer I am not a fan of Google NGrams, because they can be used without regard for proper statistical practices. It can be hard to "clean" queries enough to be useful. The phrase "close to the chest" may, in fact, be over-represented here due to medical and other bodily associations, so I would expect some blue-line inflation in the above graphs. Similarly, "close to the vest" almost certainly appears in some references to garment making and wearing. Those things aside, I believe the "close to the chest/vest" idiom undoubtedly furnishes the majority of usage instances in each case.

I have always heard/used "close to the vest", but apparently both are indeed common. The Corpus of Contemporary American English gives 89 hits for "close to (...) vest" and 36 for "close to (...) chest" when used synonymously. (There's 123 hits overall for chest, but the majority are for "she held the baby close to her chest" and other similar literal uses.)

Dictionary.com lists the origin of the phrase as mid-1900's (card-playing-based) slang, but RandomHouse's Word Maven lists uses from as far back as 1922.

another Englishman here - one who has been obsessively reading novels for a bit less than 60 years. "Close to his vest" is certainly not GB English. The first-ever time I came across "vest" was when I recently started grappling with David Baldacci's contemporary prose. It seems to be a 'word-jerk' (like knees but with words) with him and all sorts of characters quip this expression out, telling me more about the author than about the character in the book.

In every context I've ever come across it means 'secretive' - the analogy being card players who hold their cards close to their body to prevent being them being overlooked. Transferring this across the Atlantic and allowing for transliteration it's natural to suppose that as vest = waistcoat then holding your cards close to your waistcoat (as already suggested) would have led to a natural morphing of the expression.