My understanding is that no Italian word which is not a loanword can be spelled with a circumflex except for some nouns and adjectives whose plural ends in "ii". These can also be spelled with an ending "î" written in place of such "ii". In recent times, the use of such spellings, which up until recently were used as alternate spellings in some literary works for the sole purpose of making the text look more elegant, seems to have become even rarer if not almost completely nonexistent. Up until the beginning of the twentieth century, such circumflexed "î" could also be spelled as a "j". The letter "î" can be pronounced (IPA: /i/), (IPA: /ii/), or anywhere in between.
The rule for deciding whether to write the ending of a plural of a word ending in "io" as "i" or "ii" is as follows:
When a word ends in "io" and is stressed as "ìo", the plural ends in "ii" which is stressed "ìi". In this case both vowels must be pronounced separately by all Italians.
If a word ends in an unstressed "io" then the plural is always single unstressed "i" (for example, "secchi", plural of "secchio", or "vari", plural of "vario"). Using a circumflexed "î" at the end of such words would be wrong.
While the list of examples of some of the more common uses of the "î" runs into the hundreds if not the thousands, a few examples of Italian words which can optionally be spelled with a circumflex are the following (each bulleted entry given below lists all alternate spellings of an example plural form containing a circumflex in square brackets, and such spellings are listed from most frequent to least frequent):
[principi,principii,principî] (IPA: /prinˈcipi/): "principî" (circumflex version of "principii", plural of "principio", principle, pronounced "princìpi") not to be confused with "principi" (plural of "principe", prince, pronounced "prìncipi").
[geni,genii,genî] (IPA: /ˈgeni/ or /ˈgenii/): genî (circumflex version of "genii", also spelled "geni", plural of "genio", genius) not to be confused with "geni" (IPA: /ˈgeni/), plural of "gene", gene).
[assassini,assassinii,assassinî] (IPA: /assassˈinii/) (plural of assassinio, murder, not to be confused with "assassini" (IPA: /assassˈini/), plural of "assassino", murderer).
[oli,olii,olî] (IPA: /ˈoli/ or /ˈolii/): olî (circumflex version of "olii", plural of "olio", oil).
[propri,proprii,proprî] (IPA: /ˈpropri/ or /proprii): proprî (circumflex version of "proprii", plural of "proprio", belonging to one's self, of one's own).
Regards.
