The girl who they say is a singer, met me last night.
or
The girl whom they say is a singer, met me last night.
Note: We Follow British English.
The girl who they say is a singer, met me last night.
or
The girl whom they say is a singer, met me last night.
Note: We Follow British English.
You could use either as correct English.
By the way, who(m) they say is a singer is a relative clause. This means it needs parenthesis surrounding it. You have also added a capital "G" for girl. Since it is not a proper (but common) noun, it does not need one.
With the correct punctuation and grammar, it should read like this:
The girl, who they say is a singer, met me last night.
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The girl, whom they say is a singer, met me last night