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The work features a child clasping a symbol of Extinction Rebellion, the group behind the protests, crouching next to a spade and a new plant.
"From this moment despair ends and tactics begin," reads a sign next to the artwork, located at one of central London's busiest road junctions.
"I think it's virtually a certainty it's him," auctioneer Andrew Tetley told London's Evening Standard newspaper.
"It hasn't been confirmed obviously that it is Banksy yet, but clearly there's a lot of excitement," added street art fan Lindsey Solomons, 53.
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"I think the message speaks for itself really, that if we don't do something about the planet clearly we're going to be in a desperate state, and he captures that so well," she told AFP.
Environmentalists used festive but highly disruptive rallies since last week to focus global attention on climate change, bringing parts of London to a standstill.
The entire campaign saw police make 1,130 arrests and press charges against 69 people.
Banksy, whose identity is known to only a handful of friends, caused a sensation in October when one of his paintings began shredding itself, just after selling for $1.4 million (1.2 million euros).
The artist has previously espoused themes such as immigration and refugees, the Palestinian cause and the yellow vest movement in France.