Delegations from the opposing sides met on Monday and will reconvene throughout Tuesday to discuss a peaceful resolution to the war. Although no progress from the most recent talks have been revealed, Russian and Ukrainian officials gave their most upbeat assessments last weekend, prompting hopes of a positive outcome within days.
Ukraine has said it is willing to negotiate, but not to surrender or accept any ultimatums.
Ukrainian negotiator and presidential adviser Mykhailo Podolyak said: “Russia is already beginning to talk constructively.
“I think that we will achieve some results literally in a matter of days.”
RIA news agency quoted a Russian delegate, Leonid Slutsky, as saying recent talks had made “substantial progress”.
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Mr Zelensky has said he will not give into Russia’s demands and has challenged President Putin to conduct talks face-to-face.
He said that whilst this is a “hard path, this path is needed”.
Ukraine’s President suggested last weekend that future peace talks with Russia could take place in Israel, which has been engaged in diplomatic efforts to try to end the war.
Israel’s Prime Minister Naftali Bennett has held talks with President Putin and spoken by phone with Mr Zelensky.