But Kiev is especially angry at Moscow for shooting down UN attempts to condemn what Kiev views as Russia's "aggression" that led to nearly 8,000 deaths in Ukraine's separatist industrial east.
Russia portrays Ukraine's 17-month crisis as a "civil war" in which it has played no role.
The United States along with Britain and France -- the other veto-wielding Security Council members and backers of Ukraine's pro-Western leadership -- have refrained from openly backing such a drastic measure against Russia and its unpredictable President Vladimir Putin.
Ukraine's parliament Wednesday unanimously voted in favour of a recommendation for the United Nations to revoke Moscow's right to block Security Council resolutions in cases when "conflicts grow especially fierce".
Kiev's UN envoy Yuriy Serheyev tweeted that 67 of the General Assembly's 193 members have already supported the initiative.
Moscow swiftly dismissed the proposal and denounced Kiev's own push to expand veto rights to the other 10 non-permanent council members.
Ukraine hopes to be elected as one of the new rotating representatives at a vote in New York scheduled for October 15.
"We stand in favour of preserving (veto) rights for the five permanent Security Council members," Russia's RIA Novosti state news agency quoted Deputy Foreign Minister Gennady Gatilov as saying.
Gatilov called discussions about allowing the other 10 council members the right to block resolutions "premature".
- French idea -
The idea of imposing limits in extreme cases on a council members' right to torpedo legislation was initially proposed by French Foreign Minister Laurent Fabius in October 2013.
Moscow and Washington were then locked in an especially bitter standoff over Assad's alleged use of chemical arms.
US President Barack Obama was weighing the option of launching air strikes against Syria's military bases -- an idea adamantly opposed by Assad's ally Putin.
Obama's deliberations were halted by a Congress decision to prohibit direct US military involvement in the devastated Middle Eastern state.
Russia has also promised to ward off any US attempt to win the Security Council's endorsement for a military campaign in Syria.
Fabius suggested that such disputes could be resolved more smoothly if the five big powers were to "regulate their right to exercise their veto".
"The criteria for implementation would be simple: at the request of at least 50 member states, the United Nations secretary general would be called upon to determine the nature of the crime," Fabius wrote in an opinion piece for The New York Times.
"Once he had delivered his opinion, the code of conduct would immediately apply."
The initiative was later formally introduced by French President Francois Hollande and quickly supported by Mexico.
- Russian anger -
Moscow's UN envoy Vitaly Churkin told Russian reporters by video link from New York that the French plan had a "political subtext" designed to sideline Moscow.
"We believe that the right of permanent members to exercise their veto is an important mechanism that forces (countries) to search for a compromise," said Churn.
He added that Kiev's call for Western powers to expand their list of targeted sanctions against Russia over its role in Ukraine would not be discussed in New York.
"You cannot have the Security Council sit around the table and discuss sanctions," said the Russian envoy.
"Sanctions fall outside the UN Security Council mandate."