It follows criticism that HSBC, Barclays, Santander, NatWest and Lloyds are funding “fossil fuel expansion” despite making green pledges.
Businesses and charities like Greenpeace also back the campaign.
HSBC and Barclays said they were helping their clients to cut emissions.
The Make My Money Matter campaign points to research by environmental charity Rainforest Action Network, which claims that between 2016 and 2021, HSBC, Barclays, Santander, NatWest and Lloyds funnelled almost $368bn (£298bn) towards the fossil fuel industry.
It added that in the same time period, the lenders financed the 50 companies making the biggest investments in oil and gas projects to the tune of $141bn.
“HSBC was this month found to have provided $340m to a company opening a new coal mine in Germany,” it said.
The campaign, which is also backed by actor Mark Rylance and musician Brian Eno, urges the public to sign an open letter asking the banks to stop directly financing projects that expand fossil fuel use, or end relationships with clients that do.
The campaign’s founder, filmmaker Richard Curtis, said he wanted to put “a fire under the banks”.
“It’s clear that new oil and gas fields are not only hugely damaging to the planet, but they’re also wildly unpopular with the public,” he added.
Almost one third of HSBC, Barclays, Santander, NatWest and Lloyds’ customers surveyed by the campaign said that they would switch bank if they discovered that theirs was financing the expansion of fossil fuel projects.
Over 85% of customers at the five banks surveyed said they did not think that their bank was doing enough to tackle the climate crisis.
TV presenter Chris Packham CBE said that financial institutions had an “enormous ethical and moral responsibility” to start withdrawing funding from organisations that damaged the climate and biodiversity
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