Teenage climate activist Greta Thunberg, who has been nominated for a Nobel peace prize for her campaign to tackle climate change, welcomed the move and said she was "hopeful".
Members of Extinction Rebellion will call on Mr Gove to back their demands to declare a climate emergency and commit to reducing greenhouse gas emissions to zero by 2025.
A "die-in" is a demonstration in which participants lie down as if dead, and is commonly used by the group to symbolise what they say is the "sixth mass extinction".
The graffiti features an image of a child holding an Extinction Rebellion sign next to a small plant dug in the ground, with the words: "From this moment despair ends and tactics begin". We need government and business to act now.
Gove said that he agreed that emissions should be cut to zero but that "the key question" was one of "timescale" and "what is achievable".
He also said he was "open-minded" about the possibility of citizens' assemblies.
But Sam Knights, who works on Extinction Rebellion's strategy team, said the refusal to declare a climate emergency was "not only a political failure, but a moral failure".
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"I would suggest its value is well over £1 million, especially as it's classic Banksy - commentary on current political issues in a way only he can", the art dealer said.
But he added: 'He is going to meet us again in a month's time, which is a step forward.
"However, the rebellion has to continue because our demands have not been met".
The group will also meet with London Mayor Sadiq Khan on Monday and shadow chancellor John McDonnell on Tuesday.
The meeting, which lasted more than an hour, concluded with claims some progress had been made. A former Green Party county councillor, Sarah Lunnon, 54, said: 'They gave us a concrete commitment to go and talk.
Jeremy Corbyn plans to force a Commons vote on whether to declare an environmental and climate emergency.
"We know the impact climate change is having on our environment and welcome discussions on how we can tackle it".
"We want a world for those in countries worst affected by and least to blame for climate change and our young people".