Mark Carney warned he would not receive any instructions from the politicians.
In comments which risk an unprecedented clash between the Governor of the Bank of England and the Prime Minister, Mr Carney said that "it can be difficult sometimes if there are political comments on our policies".
His remarks come just days after Mrs May used her speech to the Conservative conference to criticise the Bank over quantitative easing and ultra-low interest rates.
Conservative MPs have repeatedly criticised Mr Carney over his dire economic warnings ahead of the European Union referendum.
On Friday night Tory backbenchers described his comments as "nonsense" and said that the Governor should not be "picking a fight" with the Prime Minister.
Speaking in Birmingham, Mr Carney said: "Politicians have done a very good job of setting up the system. Where it can be difficult sometimes is if there are political comments on our policies as opposed to political comments on our objectives.
"The objectives are what are set by the politicians. The policies are done by technocrats. We are not going to take instruction on our policies from the political side."
Mrs May said in her party conference speech: "While monetary policy – with super-low interest rates and quantitative easing – provided the necessary emergency medicine after the financial crash, we have to acknowledge there have been some bad side effects.
"People with assets have got richer. People without them have suffered. People with mortgages have found their debts cheaper. People with savings have found themselves poorer.
"A change has got to come. And we are going to deliver it."
The Governor also used his appearance in Birmingham to say that inflation will rise on products such as food because of the fall in the value of the pound.
In the wake of the price row between Unilever and Tesco, Mr Carney said food prices will rise and it is "going to get difficult".
Mr Carney, who was hired by former chancellor George Osborne, was vocal in warning about the risks of leaving the EU and suggested Brexit could lead to a recession.
Bernard Jenkin, the head of the Public Administration Select Committee, told the Telegraph: "The pursuit of quantitative easing is a matter for the government, which is why Mr Carney wrote to the Chancellor to ask his permission to pursue that policy.
"Monetary policy is a shared responsibility with the government and the idea that the Bank of England is completely beyond democratic accountability is a nonsense.
"Why is the bank governor picking a fight with the Prime Minister? What's his real agenda?"
Steve Baker, the Conservative MP for Wycombe, said: "These testy comments from the Governor are unwarranted.
"He and other MPC members have admitted monetary policy is redistributive so of course his work is fundamentally political."
Jacob Rees-Mogg, the Tory MP for North East Somerset, said: "It's very important that the Governor is independent.
"Unfortunately during the Brexit campaign he was not independent, he spewed forth propaganda for George Osborne's position and that I'm afraid has totally undermined his independence, to try and reassert it now is too late. It's just a pity he hasn't been independent before."
Meanwhile, it emerged that Mrs May is to visit one EU leader every five days over the next 10 weeks to press Britain's case for Brexit ahead of the crucial December EU council meeting.
Mrs May on Friday in Downing Street met all of Britain's ambassadors to the other 27 EU member states to discuss their assessment of the individual countries' attitudes to Brexit.
She told them to tell EU leaders that Britain is committed to Brexit and that getting control of UK borders is a red line in any exit negotiations.
And Michel Sapin, the French finance minister, on Friday risked anger by saying US banks have told him that they are preparing to move some operations from London to other EU cities because of Brexit.
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