The Kremlin pushed back against the idea of a face-to-face meeting involving Putin and Zelenskyy.
"First, results must be achieved through direct negotiations between the two countries," Kremlin spokesman Dmitry Peskov told reporters.
Fidan met Ukraine's Foreign Minister Andriy Sybiga in Kyiv and was due to meet Zelenskyy later in the day.
He held talks with Putin in Moscow earlier this week.
Ukraine has said it is open to further negotiations, but has not confirmed it will be in Istanbul on Monday.
At talks in Istanbul on May 16 — the first in over three years — the sides agree to swap documents outlining possible roadmaps to peace.