The most common and most likely answer to this crossword clue is the 7 letter word PITAPAT
I've seen this crossword clue in The Times
P
I
T
A
P
A
T
More clues leading to the result PITAPAT
Buddy explains!

The straight part
We know that "fluttering" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case fluttering = pitapat.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 PITAPAT mean?
PITAPAT noun- A series of quick tapping sounds
- To make a series of quick tapping sounds; to palpitate
- With a quick tapping sound