Question
What are the names of the Five villages that Krishna offers Kauravas to give Pandavas?
Answer
In this chapter of the Udyoga Parva of the Mahabharata, Yudhishthira sends an offer of peace to Duryodhana, and in the course of doing so he names the five villages:
We are desirous of peace; give us even a single province of the empire. Give us even Kusasthala, Vrikasthala, Makandi, Varanavata, and for the fifth any other that thou likest. Even this will end the quarrel. O Suyodhana, give unto thy five brothers at least five villages,--O Sanjaya, O thou of great wisdom, let there be peace between us and our cousins.
So he even left one village up to the choice of Duryodhana. In a later chapter of the Udyoga Parva, Yudhisthira laments that they weren't able to get a peaceful resolution to the conflict, and he again names the villages:
Having the Kasis, the Panchalas, the Chedis, and the Matsyas, for my allies and with thee, O slayer of Madhu, for my protector, I prayed for only five villages, etc., Avishthala, Vrikasthala, Makandi, Varanavata, with any other, O Govinda, as the fifth;--Grant us, we said, five villages or towns, O sire, where we five may dwell in union, for we do not desire the destruction of the Bharatas.--The wicked-minded son of Dhritarashtra, however, regarding the lordship of the world to be; in him, doth not agree to even that.
I assume that Avishthala is the same as Kusasthala, which according to this book is more commonly known Kanyakubja, or the modern-day city of Kannauj. As far as where the other three named villages are, this Wikipedia page provides some information as to their current location, although it doesn't cite the sources of this information:
Varanavata was the ancient capital of the Kuru kings and their forefathers. It lies on the foothills of the Himalayas, like the capital of northern Panchala viz Ahichatra. This city could be a place called Shibpuri to the north-east of Rishikesh in Uttarakhand. It was north of Hastinapura and Pramanakoti....
Vrikasthala province and town were situated in the southern part of Kuru Kingdom.... Bidding them farewell Krishna reached the town of Vrikasthala (in Gurgaon district of Hariyana) by evening and camped there that night (5,84).
Makandi was central a province running along the banks of the Ganges, south of Hastinapura. The province extended to southern Panchala Kingdom, also with the same name. Kampilya, the capital city of Panchala, was situated in the Makandi province within the southern Panchala kingdom (1,140).
Note: “The question: What are the names of the Five villages that Krishna offers Kauravas to give Pandavas?” is licensed by Stack Exchange Inc (https://hinduism.stackexchange.com/); user contributions licensed under CC BY-SA.