Bill Kidd, SNP MSP for Anniesland, lodged a motion in the Scottish Parliament informing MSPs that a convoy of 19 vehicles headed along the M74 in the early hours of Wednesday on the way to the naval base at Coulport.
His motion was backed by 25 MSPs including Glasgow SNP MSPs James Dornan, John Mason, Sandra White and Green Patrick Harvie.
Mr Kidd, co-president of the Parliamentarians for Nuclear Non-Prolifer-ation and Disarmament, said: "This practice poses an unacceptable risk to the people of Glasgow. The idea that weapons of mass destruction are being transported through our city while we sleep is absolutely chilling.
"I would urge Glasgow MSPs of all parties to join with the Scottish CND in supporting the motion in the Scottish Parliament condemning the practice, and make clear that Glaswegians should not be put in danger like this."
The MSP said the transportation from the south of England to Loch Long in Argyll was a regular occurrence.
Mr Kidd said the convoy brought weapons from the Atomic Weapons Establishment base near Aldermaston, in Hampshire, where nuclear warheads are designed and made for the UK Government.
It travelled up the M74 and through the city to the Royal Armaments depot at Coulport, where nuclear and conven-tional weapons for the Navy's submarines are housed.
It is understood that the warheads are accom-panied by a specialist safety team who are in contact with police in each region they pass through on the route.