The link, made in comments by Russia's finance minister to the Reuters news agency on Saturday, came as the Kremlin confirmed President Vladimir Putin had held private talks with Ukraine's leader Viktor Yanukovich in Sochi on Friday before the opening of the Winter Olympics.
No details of the leaders' talks were disclosed.
"We will fulfill what we have promised to Ukraine, but we would like the Ukrainian side to comply with the obligations that it has committed to," Anton Siluanov told Reuters.
The gas debt was due for repayment in late January, but Ukraine did not pay up, he said.
On Saturday, thousands of people angered by months of anti-government protests in the Ukrainian capital, Kiev, converged on one of the protesters' barricades, the Associated Press news agency reported.
They retreated after meeting sizeable resistance, averting a violent showdown.
The incident underlined tensions that persist as the protests slog through a third month with no sign of concessions from either side.
About 2,000 people streamed toward the barricade near city hall at midday, blocking traffic on the capital's main avenue and placing tires in the roadway.
Igor Polishchuk, one of the men placing the tires, said the crowd wanted to show its peaceful opposition to the protests that have pushed the country into a political crisis and complained that police had done little against the protesters.
"It's a critical mass in there, without control," he said.
"The authorities aren't anywhere inside."
EU versus Russia
The anti-government protesters have set up an extensive tent camp in Kiev's main square and occupy three nearby buildings, including the city hall, that they use for operations centres, sleeping quarters and even an improvised library.
Yanukovich has been battling massive anti-government protesters, demanding his resignation and early elections, since he rejected a trade deal with the EU in favour of closer ties with Russia.
The president must now decide whether to submit to protesters' demands by taking a more conciliatory approach towards a new agreement with the EU - a possibility that prompted Russia to suspend its bailout payments after issuing just one instalment of $3bn in December.
The US and the EU have backed the anti-government protests and promised a financial package to Ukraine on the condition that the government agrees to political reform.