Mr Hammond is said to have told the meeting that public sector workers are "overpaid" and "even a woman can drive a train".
'Later in the meeting both Boris Johnson and the PM said we should not say public sector workers are overpaid'.
Trade Minister Liam Fox, who favours making a cleaner break with the bloc, said he did not have a problem with a transition period as long as it was for a limited duration and gave Britain the freedom to negotiate its own trade deals.
The leaks expose the divisions at the heart of May's cabinet.
His outburst followed a report in The Sunday Times that he had told the weekly meeting of the Cabinet on Tuesday that public sector workers were "overpaid" when their pensions were taken into account.
In a bid to end the rows, the 1922 Committee of backbench Tory MPs told May they would back any discipline she saw fit, including firing ministers who have routinely gone off message or publicly contradicted each other since the election, according to the Daily Mail.
"I think now you would find that pretty much everybody around the Cabinet table accepts that there will be some kind of transition."
"We would not be members of the single market because we would leave the single market when we left the European Union on 29 March, 2019", he told the BBC's Andrew Marr.
And when he tried to dig himself out of the embarrassing hole, he was interrupted by Mrs May, who said: "Chancellor, I am going to take your shovel away from you". You know what everyone is desperate to get on their sun loungers and go on holiday and frankly the sooner they do the better. First secretary Damian Green said: 'Now is absolutely not the time for this type of activity'.
JP: "Put the record straight for us then".
Number 10's official spokesman said there would not be a "formal investigation" into the leaks from the Cabinet meeting, but Mrs May would be "reminding" ministers of their "responsibilities".
Christina McAnea, Unison assistant general secretary, commented: "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".
After the protest they will submit a petition to the Treasury protesting at falling public-sector wages. "What is key is we have to strike the right balance between being fair in pay to public sector workers and fair to tax payers to ensure we still have a strong economy so we can employ and pay those public sector workers".
A "frustrated" Chancellor warned his cabinet colleagues he was "determined to keep a tight rein on public borrowing and that taxes may have to rise to fund new spending commitments", adds the paper.