"The Russian side has made the step to continue talks, which have been very intensive over the last few days," Alexei Miller told state-owned Rossiya 24 television from Brussels.
"The Russian side has decided to move the introduction of pre-payment (to June 16) at 10 am (2.00 am ET)."
Gazprom was not immediately available to comment, reuters reported.
Russia, Ukraine and the European Union will resume efforts to resolve the gas pricing dispute on Wednesday after an earlier Russian deadline for Kiev to pay some of its debts passed without Moscow cutting off supplies.
Ukraine wants to change the 2009 contract that locked it into buying a set volume of gas, whether it needs it or not, at $485 per 1,000 cubic meters -the highest price paid by any customer in Europe.