Orban hinted that these meetings could take place next week but did not provide further details.
Meanwhile, Hungarian Foreign Minister Peter Szijjarto announced upcoming Ukrainian "peace missions" next week.
In a Facebook post, Szijjarto addressed criticism from European politicians over Orban's meeting with Putin.
"These attacks will not deter or embarrass us. The mission of peace continues and is even getting stronger, so I ask European politicians who support the war to fasten their seatbelts and stay tuned for updates next week," Szijjarto said.
Orban reiterated his commitment to resolving Russia's war against Ukraine on July 8.
"I have concluded my talks in Moscow with President Putin. My goal was to open channels of direct communication and start a dialogue on the shortest road to peace. Mission accomplished! To be continued on Monday," he wrote on X on July 5 after his official visit to Moscow.
Earlier, on July 2, Orban made his first visit to Kyiv in 12 years, meeting with President Volodymyr Zelenskyy and proposing a "ceasefire" to accelerate negotiations with Russia.
Orban suggested that Kyiv's peace initiatives "take a lot of time" and asked Zelenskyy to consider whether it would be possible to "first, cease fire, and then negotiate peace."
"A ceasefire could accelerate the pace of these negotiations," Orban said.
In response, Deputy Head of the Presidential Office Ihor Zhovkva stated that "Hungary is not the first country to talk about such a scenario." Zhovkva clarified that Zelenskyy rejected such proposals: "He listened to Orban, but in response, he stated his position, which is clear and known to everyone."
"Such processes cannot be perceived in isolation," Zhovkva said. "Yes, indeed, Ukraine wants peace, and we have a tool for this - a peace summit."
Orban is known for his pro-Russian and anti-Ukrainian statements. He has called Ukraine a "non-sovereign state," claimed the EU lacks funds to support Ukraine, and defended Putin while calling for negotiations with him.