The Canadian will not take up his option to leave in 2018, but has declined to serve the full eight-year term that would have seen him stay on at the central bank until 2021.
"I would be honoured to extend my time of service as governor for an additional year to the end of June 2019," he wrote in a letter to finance minister Philip Hammond.
Prime Minister Theresa May is expected to trigger Article 50 of the EU's Lisbon Treaty by the end of March, starting a two-year exit process that would see Britain leave the European Union by early 2019.
Carney has been strongly criticised by members of May's Conservative party over his warnings about the impact of Brexit before the June referendum, while there was speculation about tensions with the government over policy.
More From This Section
"This will enable you to continue your highly effective leadership of the bank through a critical period for the British economy as we negotiate our exit from the European Union," Hammond wrote.
"I am grateful for your contribution to both monetary and financial stability to date, and I look forward to your continuing contribution in the future."
Carney began his tenure in 2013, stressing his intention to serve only five years of an eight-year term for personal reasons, meaning he would step down in 2018.
Amid intense speculation in the media over the weekend, May offered her full support today morning.
Her spokeswoman said Carney was "absolutely" the right person for the job, adding: "It's clearly a decision for him but the prime minister would certainly be supportive of him going on beyond his five years."
"The prime minister has always had a good working relationship with the governor of the Bank of England and intends to continue that," she said.
The Sunday Times, citing senior figures who had worked closely with the governor, had reported that he was "disillusioned" with the government and criticism from sections of her party.