The Italian consulate in Bucharest have an astonishingly awful website which fails to list any useful information. However their consulate in the US has a good website, which lists the following question in their FAQ:
My US visa is expired as well as my I-94 and I cannot renew it in the US. Can I apply for a Schengen visa?
No, you cannot. We can process visa request only from legal residents of the US.
I presume their Romanian office has a similar policy, so you wouldn’t be able to apply there.
Credit:stackoverflow.com‘