Responding to news of the plans, backbench Labour MP Rachael Maskell said she felt uneasy about more North Sea drilling and said at a time when the UK was experiencing heatwaves and wildfires it was more important than ever to ” follow the climate science because we have got a climate emergency”.
“The climate science is really clear we cannot continue to use carbon based fuels,” she added.
Earlier this week, before the leadership nominations deadline, Burnham was issued with a letter from the oil and gas industry and trade unions calling on him – and all other Labour MPs – to “back North Sea oil and gas”.
It stated that support for the oil and gas industry is “a signal that the country remains committed to producing, building and manufacturing.
“It is a signal that government backs the people and places that have powered this country for generations,” the letter added.
But the suggestion of new plans has already drawn criticism from the Green Party, with MP Adrian Ramsay calling approval of new drilling “the wrong response” and “will do nothing to bring down bills”.
“Andy Burnham says he takes the climate and nature crises seriously, but words are no substitute for action,” he said.
“The science is clear: if we are serious about limiting climate breakdown, we cannot keep opening up new fossil fuel projects,” he added.
Burnham, who returned to Parliament a month ago in a by-election, emerged as the sole leadership candidate after being backed by 379 Labour MPs, as well as all 11 trade unions affiliated to the party, earlier this week.
The new Labour leader has said he is finalising his cabinet before he takes over from Sir Keir.
There has been growing speculation over who will take the top jobs around Burnham, with Miliband and Shabana Mahmood thought to be in the running to replace Rachel Reeves as chancellor.
Asked why he had not announced his top team, the new Labour leader said on Friday: “It would be somewhat premature and would, I think, cause complete chaos if you start half a reshuffle before you’re in the position.”
Earlier that day, in his first speech since receiving the party backing, Burnham promised he has “a plan” and vowed to “bring back hope” under his leadership.
The address also offered the first glimpse of Burnham’s policy agenda – which is said to include new council homes, reinvigorated high streets and improvements to education.
The former mayor of Greater Manchester has also previously outlined his vision for reforms to social care, greater public control of water and energy firms and the devolution of power away from Whitehall.