Russian news agencies quoted Kremlin spokesman Dmitry Peskov as neither confirming nor denying the Friday visit.
State news agency RIA cited Peskov as saying that Putin had a "busy schedule" on Friday, about which the Kremlin would inform reporters later.
It would be the first time since Russia's invasion of Ukraine that Putin and Orban have met inside Russia.
The road to — Orbán Viktor (@PM_ViktorOrban) #Kyiv. A trip for #peace. pic.twitter.com/nWFf0g86EJJuly 3, 2024
Orban, who has maintained a closer relationship to Russia than other European Union member leaders since the invasion, visited Russia in 2022 without meeting Putin and has met him in other countries.
From the start of this month, Hungary has taken over the rotating presidency of the EU, a largely ceremonial role tasked with agenda-setting and brokering agreements in legislative affairs.
Earlier this week, Orban visited Kyiv where he urged Ukrainian President Volodymyr Zelenskiy to consider a ceasefire to accelerate an end to the war with Russia.
"The EU rotating presidency has no mandate to engage with Russia on behalf of the EU," Charles Michel, the president of the council of European Union leaders said on social media platform X reacting to Orban's visit to Moscow.
"No discussions about Ukraine can take place without Ukraine," he added.
Hungary, a member of the EU and the NATO military alliance, has refused to send weapons to Ukraine and vocally criticised EU sanctions against Russia while stopping short of using its veto power to block them.
Ukrainian local authorities said Russian strikes killed two people and wounded 26 on Thursday in regions of the country stretching from the south to the east and northeast.
A missile strike in southern Odesa region killed a woman, injured seven people and damaged port infrastructure, regional governor Oleh Kiper said on Telegram.
In northeastern Kharkiv region, a second woman was killed and a man wounded in a strike by a Russian guided bomb on the village of Ruska Lozova, according to regional governor Oleh Syniehubov.
Nine others, including four children, were wounded in a drone attack and shelling in the town of Novohrodivka, in the frontline Donetsk region, governor Vadym Filashkin said.
Dnipropetrovsk regional governor Serhiy Lysak reported seven wounded in the southern town of Nikopol.
He had said earlier that Russian forces had attacked areas near Nikopol with drones and artillery on Wednesday evening and Thursday morning.
Lysak later reported that a woman died of injuries sustained on Wednesday in the region's main city, Dnipro.
Seven people died in that attack.
Two civilians in the southern city of Kherson were wounded in a drone strike, the local administration said.
All the affected regions have been subjected to repeated attacks since the start of Russia's full-scale invasion in February 2022.
Russia denies targeting civilians or civilian infrastructure but thousands of people have been killed and wounded.
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