The National Citizen Service (NCS) is looking for a creative agency to carry out the rebrand on a contract worth between £4m and £10m over the next two years.
Between 2010/11 and 2017/18 funding for youth services has been cut from £652m to £352m.
However, between 2014/15 and 2017/18 the government spent £634m, which is 95 per cent of its youth services budget, on the NCS, which offers summer activities to 15- to 17-year-olds.
The LGA says NCS money would be better spent on year-round provision for young people of all ages.
The council representative body also points out that the NCS rebrand cost could be more than the £9.8m pledged by the government last month to tackle knife violence by helping community groups turn young people away from crime and gangs.
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"It is not right that £10m of government money is to be spent on a brand refresh for the National Citizen Service," said LGA children and young people board chair Anntoinette Bramble.
"This sends the wrong message at a time when councils are being forced to cull vital youth services as a result of government funding cuts.
"We have long called for some of the funding for the NCS to be devolved to councils to fund youth services.
"While we recognise that NCS can be a positive experience for those who take part, funding all-year-round youth services in local communities would be far more effective in reaching out to young people and helping them to thrive and prosper.
"The shocking surge in children and young people tragically involved in knife crime and gang activity only reinforces the need to invest in local youth services to support and work with children and young people to help them stay safe."
The rebranding tender for the NCS says it is looking to "inspire more young people to participate in the programme".
It adds: "We are looking for an agency who can help us deliver a razor sharp brand proposition, a brand identity that is fit for purpose in the digital age and a creative platform that can not only cut through the clutter but live and breathe through every single touchpoint in our customer journey to make NCS the best known and loved youth movement brand in the UK."
Figures for the NCS in 2016 found that only 12 per cent (93,000) of eligible young people took part in the programme.
The NCS Trust, which runs the NCS, denied that the brand refresh spend will be as high as £10m.
A spokeswoman said: "To be clear, £10m will not be spent on any brand refresh by the National Citizen Service (NCS).
"The figure will be around £1m each year for our ongoing marketing and creative services, including a brand refresh, to drive participation and inspire young people to take part in the NCS programme.
"'£10m' was stated in a procurement document as an upper limit and no government funding will be spent on the brand refresh element of this contract."
The trust is ambitious to expand its reach, with 500,000 young people having taken part in NCS, and it is "necessary to market a national programme of this size", added the spokeswoman.
She continued: "NCS Trust wholeheartedly agrees that local youth services must see a strong surge in funding and believes youth services should be statutory."