"He is too removed from the reality of daily life to see the impact of seven years of pay pinching on ambulance workers, nurses and teaching assistants", said GMB national secretary Rehana Azam.
A Treasury source insists he did not use the word "overpaid".
Asked directly whether he felt public sector workers were overpaid, Mr Hammond said: "This is a relative question; this is about the relationship between public and private sector pay".
John McDonnell slammed the Chancellor for the comments, allegedly made during a cabinet meeting last week.
Liam Fox, the global trade secretary, who is seen as among the hardline Brexiteers, condemned the leaks, which, he argued, would only give succour to Brussels.
"Some of the noise is generated by people who are not happy with the agenda I've tried to advance over the past few weeks", Mr Hammond said.
Mr Hammond said that government ministers were becoming increasingly convinced of the need for transitional arrangements to reduce disruption as the United Kingdom leaves the EU.
Milano, poliziotto ferito in stazione Centrale: era intervenuto per lite su pullman
Un agente della Polizia è stato ferito con un arma da taglio durante un controllo alla Stazione Centrale di Milano . L'agente, a quanto si è appreso, è stato protetto dal giubbotto antiproiettile e ha riportato una ferita lievissima.
Ministers have talked of a transitional period after the United Kingdom leaves the European Union to avoid a "cliff edge" scenario, but the length and precise arrangements have yet to be decided.
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.
Mr Hammond said: "Public sector pay raced ahead of private sector pay after the crash in 2008-09".
DG: "Every July Westminster gets feverish and every July people say it's different this year and you know what it's the same every year".
"And I don't for a moment deny that there are areas in the public service where recruitment and retention is becoming an issue, that there are areas of the country where public sector wages and private sector wages are getting out of kilter in the other direction and we have to look at these things and we have to discuss them".
Iain Duncan Smith, former Tory leader, said he was dismayed at the leadership speculation in the Sunday papers, telling some of his colleagues: "Just for once, shut up for God's sake".
"I'm not going to get into speculation of who said what where and when".
The minister claimed Mr Hammond views Brexiteers as "a bunch of smarmy pirates" who have "taken the Establishment prisoner".