"There can be ups and downs in the two countries' relations (but) we need to remove elements that harm our relations," Yildirim told journalists in Istanbul, referring to the Pennsylvania-based cleric.
Ankara has for years accused Gulen of running a "parallel state" in Turkey and it also blamed him for ordering the failed putsch of July 15.
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The reclusive cleric, who has lived in self-exile in the United States since 1999, strongly denies any involvement in the coup attempt.
Ankara had previously suggested any failure to deliver Gulen would severely damage bilateral ties and Justice Minister Bekir Bozdag said it was up to Washington to extradite him to prevent "anti-US feeling" in Turkey turning into "hate".
The White House has confirmed that US Vice President Joe Biden will visit Ankara next week in the highest-ranking visit to Turkey by any Western official since the coup.
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Yildirim also confirmed a technical delegation would arrive on August 22 for talks with Turkey's judicial authorities ahead of Biden's visit.
"I hope this process will be finalised in a way favoured by both countries, and (that) questions in the minds of Turkish people about America will be removed."
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