The Ottoman Empire probably had the best claim as an extension to the Roman Empire, and Mehmet II as far as I know styles himself “Emperor of the Romans”, but with Suleiman the Magnificent the empire became much more “Muslim”. I was wondering how strongly the Ottomans felt they were an extension of the Roman Empire, or maybe even the Caliphate?
Not to discourage others giving an answer, but quite a similar question was asked only a week ago, with contributions by myself, /u/Sinan-Pasha96, and /u/Chamboz. Again I invite others to provide a more comprehensive answer as I cannot right now, but put bluntly, somewhat, but not particularly.