The most common and most likely answer to this crossword clue is the 7 letter word WHISPER
I've seen this crossword clue in Independent
W
H
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S
P
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R
More clues leading to the result WHISPER
Buddy explains!

The straight part
We know that "rumour" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case rumour = whisper.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 WHISPER mean?
WHISPER noun- The act of speaking in a quiet voice, especially, without vibration of the vocal cords.
- (usually in the plural) A rumor.
- A faint trace or hint (of something).
- A low rustling sound, like that of the wind in leaves.
- A private message to an individual in a chat room.
- To speak softly, or under the breath, so as to be heard only by one near at hand; to utter words without sonant breath; to talk without that vibration in the larynx which gives sonorous, or vocal, sound.
- To mention privately and confidentially, or in a whisper.
- To make a low, sibilant sound.
- To speak with suspicion or timorous caution; to converse in whispers, as in secret plotting.
- To address in a whisper, or low voice.
- To prompt secretly or cautiously; to inform privately.