The most common and most likely answer to this crossword clue is the 4 letter word PAPS
I've seen this crossword clue in The Guardian
P
A
P
S
Buddy explains!

The straight part
We know that "breasts" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case breasts = paps.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 PAPS mean?
PAPS noun- Food in the form of a soft paste, often a porridge, especially as given to very young children.
- Nonsense.
- Porridge.
- Support from official patronage.
- The pulp of fruit.
- To feed with pap.
- A female breast or nipple.
- A man's breast.
- A rounded, nipple-like hill or peak.
- Pap smear
- A paparazzo.
- (usually in the passive) Of a paparazzo, to take a surreptitious photograph of (someone, especially a celebrity) without their consent.
- Pa; father