According to Shia Muslims the only current legitimate Caliph selected by Allah is Imam Mahdi a.s. but he is now at occultation. so because Muslims recognize him as Caliph so no Shia Muslims claim this title. Shia Muslims believe according to different verses of Quran only Allah has the right to select his own Caliph on earth. because Caliph means deputy and representative of Allah in earth and only Allah can select his own representative.
And [mention, O Muhammad], when your Lord said to the angels, "Indeed,
I will make upon the earth a Caliph." They said, "Will
You place upon it one who causes corruption therein and sheds blood,
while we declare Your praise and sanctify You?" Allah said, "Indeed, I
know that which you do not know." http://tanzil.net/#2:30
In this verse God says "I" select Caliph.
Also other verses say God selects Caliph.
Shia Mulslims believe during occultation times Wali Faqih undertakes the duties of Caliph. the current Wali Faqih of shia Muslims is Imam Khamenei.
Reference and more study:
Islamic Government: Governance of the Jurist