Ukraine replied that although it agreed in principle to paying market prices for gas, it wanted clearer guidance on exactly what figures Russia had in mind.
Moscow had said it would cut off gas supplies to Ukraine from 0700 GMT on 1 January if there is no agreement, sparking fears that deliveries to snowbound Europe could be affected.
Putin told gas monopoly Gazprom on Saturday to supply Ukraine at 2005 prices for the first three months of next year if Kiev signed a new contract agreeing to "market prices" from April.
"Above all Ukraine is a brother nation and we must think about the whole relationship between Russia and Ukraine," Putin told a televised meeting of the Security Council at which the head of Gazprom, Alexei Miller, was present.
Gazprom wanted Ukraine, led by Viktor Yushchenko, the pro-Western president, since last year's "Orange Revolution", to agree to steep increases in prices for its gas - to $230 per 1000 cubic metres from the current $50.
"Should there be no clear reply, we will consider that our offer has been rejected," Putin added.
What is market price?
Ukrainian officials accuse Moscow of using the issue to punish Kiev for its drive to join both the European Union and NATO a year after mass protests helped propel Yushchenko to an election victory - beating a Kremlin-backed candidate.
Yushchenko: Gazprom is adjusting
prices according to customer
"As for the statement by Russia's leadership, we accept the proposal on a transition to market prices for gas," Interfax Ukraine news agency quoted Valentin Mondrievsky, spokesman for Ukraine's prime minister, as saying.
"But there must be talks on figures and both sides must agree their positions. We have to fix a day for sitting down and continuing talks."
A statement on Yushchenko's website (www.president.gov.ua) said he believed the two sides were ready to negotiate, but made no reference to the Kremlin proposal.
Yushchenko, who has described $80 as a fair price, has said Gazprom was adjusting prices according to the customer - $120 for the Baltic states, $110 for ex-Soviet Caucasus states and $47 for pro-Russian Belarus.
Analysts said Ukraine's leaders should have launched serious talks far sooner, but had no doubts about Moscow's motives.
"It is clear Russian authorities are using Gazprom to solve several issues with Ukraine"
Comment in Ukranian weekly Zerkalo Nedeli
"It is clear Russian authorities are using Gazprom to solve several issues with Ukraine," Ukrainian weekly Zerkalo Nedeli wrote.
"Key among them is a drive to confine Ukraine to a channel and kill off once and for all any desire to look towards Europe or even dream of having our own foreign and economic policies."
While maintaining pressure on Ukraine, Gazprom assured Europeans that any shutoff could be selective and they would not go short.
But Italian oil and gas firm Eni said it had been warned by Gazprom that supplies could be disrupted. Central European states set up contingency plans. Poland said it had at least a week's reserves of gas to guard against supply interruptions.
The EU has called a 4 January meeting of energy officials from its member states to work out a common approach. So far it has stayed out of the row, saying it is up to the two neighbours to negotiate a settlement.