Barnier proposed the move, which would also buy May more time to strike a trade deal with Brussels.
Britain would remain tied to the EU until the end of 2021, rather than December 31, 2020.
Tory Eurosceptics reacted with fury, saying the move would add up to £17 billion ($31 b) to the Brexit bill and could cost the Conservatives the next election because the benefits of leaving the EU would not have been felt by 2022.
The DUP said the time had come to "show the EU negotiators the door".
Philip Hammond, the Chancellor, in effect confirmed an extension of the transition period saying: "It is true that there needs to be a period - probably following the transition period that we've negotiated and before we enter into our long-term partnership - just because of the time it will take to implement the systems required."
British negotiators believe an agreement to extend the transition period would help engineer a breakthrough in the Brexit negotiations by pushing back the need for a "backstop" solution to the Irish border problem.
May is also trying to fend off the threat of Cabinet resignations after at least four senior ministers expressed concerns about her current backstop, that would involve the whole of the UK staying in a customs union with the EU with no specified end date.
Dominic Raab, the Brexit Secretary, raised the stakes on Saturday by saying that any customs union backstop that was not "finite" would fail to deliver on the result of the EU referendum.
On Sunday, 63 Eurosceptic Tory MPs issued an open letter in which they challenged the government's proposal.
On Saturday, No 10 insisted that May would never agree a deal "that would trap us in a backstop permanently", but refused to guarantee it would be time-limited.
Extending the transition period would also confront the reality that trade experts have no confidence that a EU-UK trade deal can be completed in the 21 months currently available.
However, Britain would almost certainly have to pay billions more for continued access to the customs union and single market during an extended transition, as the £39 billion Brexit "divorce" bill only covers the financial period up to 2020, when the EU's current seven-year budget cycle expires.
Brexiteers fear the UK being trapped in a never-ending "limbo" Brexit.
Jacob Rees-Mogg, the leading Eurosceptic MP, warned that extending the transition period would be a "very high-risk strategy" with the UK "liable for our share of the bills without having a say in the budget".
He added: "It not only leaves us in the EU for longer, it leaves us as a non-voting member for longer so we have no say over the rules coming in.
"It's also bad news from a party point of view because we would not have been able to show any of the advantages of leaving the EU by the time of the next general election."
It is not clear whether EU leaders will agree to such a deal when they meet in Brussels next week.
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Despite May's insistence that any backstop plan would be temporary, Simon Coveney, Ireland's deputy prime minister, said imposing a time limit would be a "deal breaker".
However, Raab said: "It would have to be finite, it would have to be short and it would have to be, I think, time-limited in order for it to be supported here. What we cannot do is see the UK locked in via the back door to a customs union arrangement which would leave us in an indefinite limbo. That would not be leaving the EU."
Sammy Wilson, the DUP's Brexit spokesman, told May: "Go with your principles, go with your instinct, go with your responsibility to the people of the UK and show the EU negotiators the door."
Telegraph, London; Reuters