The councils estimated the costs at around £4.6 million, between 2010 and 2011.
But the LGA has warned MPs that if fewer than 102 of them attend the vote, the Bill will fail and action to tackle metal theft would stall for at least another year.
Cllr Mehboob Khan, Chairman of the LGA's Safer and Stronger Communities Board, said: "This mindless crime is spiralling out of control and has cost councils millions of pounds in having to replace memorial plaques, manhole covers, metal gullies, children's playground equipment, street signs and lead from schools, council offices and crematoriums.
"MPs will all be aware of the damage, disruption and heartaches metal thefts have caused in their areas. The legislation for regulating the scrap metal industry has long ceased to be fit for purpose and we can't afford for Parliament to stall any longer on bringing it up to date.
"Whatever their political party, councils from across the country will be looking to MPs this Friday to take forward this Bill to help introduce proper regulation of the scrapyards where much of this stolen metal ends up."
Gully covers were the most frequently stolen item, with 31.8 per cent of countils reporting their theft. Other common targets were roofing materials (24.2 per cent), road signs (14.6 per cent) and electric cables (14.6 per cent)
One in five councils said the thefts had led to 'major consequences' for residents - such as a road being closed or a member of the public falling into a drain.
Cllr Khan added: "Soaring thefts of metal are causing increasing amounts of disruption and distress, whether it's homes left without power because of stolen cabling or bereaved families heartbroken at the theft of a plaque commemorating the heroic efforts of a loved one.
"We simply can't afford to do nothing, but frustratingly, councils are being hindered by out of date legislation which hampers their ability to properly regulate the scrap metal industry.
"Councils want to see a renewable licence for scrap metal dealers, which could be reviewed at the instigation of the police or licensing authority and, if necessary, revoked.
"This would make it much less likely that stolen metal will find its ways into scrapyards."