"We do not need mediation ... for Israel in any way," Davutoglu said during a televised press conference in the central province of Konya when asked to comment on the possibility of the US helping to resolve their differences.
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"There is no such situation in which mediation is needed. The demands of Turkey are clear" if its former ally Israel wants to improve relations, Davutoglu said.
"No one should test our resolve on this matter," he said, adding that Israeli-Turkish relations might be on the agenda among other issues of a meeting between Turkish Prime Minister Recep Tayyip Erdogan and US President Barack Obama next week on the sidelines of the UN general assembly.
"The Americans are probably the people who best understand Turkey's position on this issue," Davutoglu added.
Addressing Cyprus' plan to begin gas exploration in the Mediterranean, Davutoglu said such a move would amount to a "provocation" and if drilling went ahead Turkey would consider carrying out its own offshore surveys with northern Cyprus.
"If this fait accompli continues we have steps of our own that we will take ... northern Cyprus can carry out the same explorations with Turkey and TPAO," he said, referring to Turkey's state-owned oil and gas exploration company.
Earlier this week, Erdogan continued his aggressive rhetoric towards Israel, saying the Jewish State could not do "whatever it wanted" in the eastern Mediterranean and that Turkish warships could be there at any moment.
Erdogan also spoke of the diplomatic crisis between Israel and Turkey, saying that "Islam and democracy are not contradictory" and reiterating that the only way to normalize the situation is for Israel to offer an official apology for the 2010 raid on the Gaza-bound aid flotilla.
Reuters contributed to this report
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