Former chancellor Mr Hunt came under fire from environmental lobby groups last year when he downgraded climate change, mentioning it only once in his letter.
Before taking power, Labour vowed in its manifesto that it would reverse the decision to stop the Bank from giving “due consideration” to climate change in its mandate.
However, Ms Reeves’s move to reintroduce a remit to combat risks from climate change could prove controversial.
Lord King told The Telegraph last year that it made “absolutely no sense” to make net zero another responsibility for the Bank.
“The Bank of England can do nothing about climate change,” he said, adding that the institution should focus on interest rates and keeping prices in shops stable.
There have been other warnings, with Mr Bailey conceding that while there was no excuse for failing to tackle climate change, it was not the Bank’s job to help “the world adjust”.
Similarly, the Economic Affairs Committee in the House of Lords said last year that the central bank’s focus on net zero “jeopardises” its ability to face down inflation.
The Committee urged the Treasury to “prune” the Bank’s responsibilities and said it risked becoming politicised because of avalanches of priorities it was being asked to consider.
The Government has made the environment a key policy issue, with its election manifesto promising to make Britain a “clean energy superpower” and accelerate the transition to net zero.
Ms Reeves has also enlisted former governor Mark Carney, who made climate risks a defining issue during his time in office, to advise on unlocking private investment and creating a national wealth fund.