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Mr Kalin, who also serves as special adviser to President Recep Tayyip Erdogan, was accompanied by Deputy Foreign Minister Sedat Onal, the Turkish presidential office said.
A statement on the Turkish presidency's website gave no details about the meeting but Mr Kalin's visit comes as Turkey tries to mediate between Ukraine and Russia in an effort to end the war.
Turkey hosted a meeting between Moscow and Kyiv negotiators in Istanbul and another between Russian Foreign Minister Sergey Lavrov and his Ukrainian counterpart Dmytro Kuleba in Antalya in March. However, the talks stalled after the discovery of mass civilian casualties Bucha, near Kyiv, after Russian troops pulled back from an attempt to seize the Ukrainian capital.
Mr Erdogan is now trying to pave the way for an Istanbul summit between Russian President Vladimir Putin and Mr Zelenskyy.
Lower-level negotiations between Russia and Ukraine have continued throughout the conflict but without any breakthrough.
Mr Putin told UN chief Antonio Guterres in Moscow that he still held out hope for negotiations to end the war, which Russia calls a "special military operation".
"Despite the fact that the military operation is ongoing, we still hope that we will be able to reach agreements on the diplomatic track," Mr Putin told Mr Guterres in Moscow, in televised remarks.
"We are negotiating, we do not reject [talks]."
Ukraine has proposed an international agreement whereby other countries would guarantee its security. In return, Kyiv would not join Nato or host foreign military bases.