The most common and most likely answer to this crossword clue is the 6 letter word SCAMEL
S
C
A
M
E
L
Buddy explains!
The straight part
We know that "Bird in Shakespeare supposedly" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case bird in shakespeare supposedly = scamel.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.