According to media reports, during a discussion on transport, the chancellor quipped that driving trains had now been made so easy that "even a woman" could do it. Ms May rebuked him after his remark, The Sun reported.
"The care worker hurrying from house to house doesn't feel overpaid, nor does the hospital cleaner working round the clock, or the teaching assistant going the extra mile for the children she supports", she said.
The paper said it had five sources for the story, the latest in a series of hostile briefings against the Chancellor, which threatened to fuel growing public anger over the Government's continuing 1% public sector pay cap.
He told the BBC's Andrew Marr Show: 'Some of the noise is generated by people who are not happy with the agenda which I, over the last few weeks, have tried to advance'.
"I think most people are wiling to accept a transition, so long as it's of a limited duration in order to avoid a hard landing", he said.
Mr Hammond believes government ministers were becoming increasingly in favour of the need for transitional arrangements to reduce disruption during the process of Britain leaving the EU.More news: First look at Emily Blunt as Mary Poppins
Former party leader Ian Duncan Smith told the BBC there was no appetite among conservative lawmakers for a leadership contest and said his colleagues should "shut up" and "let everyone else get on with the business of governing". Not just Brexit, but acid attacks we need to take action on this, terrorism....
Public-sectors workers will be joined by Ms O'Grady at a protest at Victoria Tower Gardens, outside Parliament, at 10.15am today.
Piling on the pressure, a report by the TUC published on Monday suggests that public sector workers are now thousands of pounds worse off since the Conservatives came to power in 2010, with paramedics and prison officers out of pocket by up to £3,800 (US$4,960) annually.
Unison has also come out strongly against the Tory chancellor's remarks, as it said that he was "completely out of touch" with the difficulty faced by many employed in the sector.
Brexit minister David Davis is due in Brussels on Monday for the next round of talks with European Union officials.
Asked whether Mrs May rejected the unnamed minister's claim that the Treasury was trying to undermine Brexit, the spokesman said: "I'm not getting into anonymous quotes".
The minister claimed Mr Hammond views Brexiteers as "a bunch of smarmy pirates" who have "taken the Establishment prisoner".