The most common and most likely answer to this crossword clue is the 6 letter word BOSOMS
B
O
S
O
M
S
More clues leading to the result BOSOMS
Buddy explains!

The straight part
We know that "chests" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case chests = bosoms.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does BOSOMS mean?
BOSOMS noun- (somewhat obsolete) The breast or chest of a human (or sometimes of another animal).
- The seat of one's inner thoughts, feelings etc.; one's secret feelings; desire.
- The protected interior or inner part of something; the area enclosed as by an embrace.
- The part of a dress etc. covering the chest; a neckline.
- A woman's breast(s).
- Any thing or place resembling the breast; a supporting surface; an inner recess; the interior.
- A depression round the eye of a millstone.