The agreement could potentially see the NHS save millions if it goes through - but the Conservatives say that talks should be scrapped in light of the recent deterioration of relations between the two countries.
Tensions have risen to new heights in recent days after several recent incursions into British aerospace by Russian bombers.
"It seems ludicrous to me that we are even discussing allowing Gazprom to bid to supply British hospitals," shadow health minister Mike Penning said. "We cannot afford to put the energy supplies to our hospitals at risk. This is an area where if the heating supply goes off lives could be put at risk.
"Whatever suppliers we are looking at have to be able to guarantee security of supply and I do not think Gazprom can do that because they are in the hands of their political masters in the Kremlin."
The Russian parliament recently voted to allow Gazprom to raise its own army.
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