While a number of EU teams are already in position, others are not, and Russian monitors have still to deploy to stations in western Ukraine, our correspondent says.
Moscow turned off the taps last week after it accused Kiev of stealing gas meant for other European customers.
Alexander Medvedev, deputy chief executive of Russian gas giant Gazprom, told a news conference in Brussels on Monday: "If there are no obstacles... gas supplies will be restarted at 8 o'clock [0700 GMT].
"[We] will all hope it will happen tomorrow."
Earlier, Ukraine had dropped provisos it wanted to add to the agreement.
Russian President Dmitry Medvedev had emphasised that Russian gas to Europe via Ukraine would flow only when monitors were in place and a transit deal had been signed by all parties.
Russia cut gas supplies to Ukraine on New Year's Day, saying it would pump only enough for customers further down the pipeline.
But then Moscow accused Ukraine of siphoning off gas intended for third countries and it restricted supplies even further.
Ukraine denied the claim, but the flow of Russian gas ceased completely on 7 January, leaving many European countries with major shortages.
The EU called the supply cut "completely unacceptable", and entered into shuttle diplomacy between Kiev and Moscow.
A deal was struck at the end of last week, but fell through when Moscow alleged that Ukraine was trying to deny its debt to Russia for gas supplies.
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