"As soon as people show up there and really sit down and start working, gas will immediately resume flowing," he told reporters at his residence outside Moscow.
Ukrainian officials say the monitors could be in place as early as Friday.
Mr Putin also said Russia was prepared to pay the market price for Ukraine's transit fee, as long as Kiev paid the market price for gas for its own use.
However, Russian officials say they have not yet signed a deal with Ukraine.
Oleg Dubyna, the head of Ukrainian gas company Naftogaz, said Kiev would transport Russian gas to Europe as long as Russia provided additional gas to keep the system running.
Some EU states are getting no gas at all or have seen supplies sharply cut.
Ukraine denies Russian accusations that it is stealing gas passing through export pipelines on its territory.
Russia cut gas to Ukraine itself a week ago as the row over pricing and allegedly unpaid bills escalated.
The EU depends on Russia for about a quarter of its total gas supplies, some 80% of which are pumped via Ukraine.
Brussels has so far avoided taking sides in the dispute, calling only for deliveries to resume urgently, the BBC's Gabriel Gatehouse reports from Kiev.
Apart from the immediate question of who is to blame, Kiev and Moscow have fundamental disagreements over how much Ukraine owes Russia for last year's gas and how much it should pay this year, our correspondent adds.
On Wednesday, heating systems shut down in some parts of central Europe, as outdoor temperatures plunged to -10C or lower.
The list of countries that reported a total halt of Russian supplies via Ukraine included Romania, the Czech Republic, Slovakia, Bosnia-Hercegovina, Bulgaria, Croatia, Greece, Hungary, Macedonia, Serbia, and Austria.
Countries in eastern and central Europe have been particularly badly affected, as they rely heavily on Russian gas supplies but don't have access to the same kind of reserves found in Germany, Italy and France.
Power stations have been told to switch to fuel oil where possible, while big industrial users have been told to prepare to limit or halt use.
There have also been calls for Soviet-era nuclear plants to be restarted in Bulgaria and Slovakia.
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