Staying in the customs union would let companies in the United Kingdom continue to trade freely in the EU.
Parliament now "deserves a say", he said, adding that there was "perhaps an opportunity to consult more widely with the other parties on how best we can achieve it".
Besides forging a deal to keep her job and preparing for Brexit talks, May is also grappling with a crisis over the breakdown in power-sharing between the pro-British DUP and the Catholic nationalist Sinn Fein party in Northern Ireland's separate, devolved government.
"We are just about to start the negotiation".
"We have been very clear that we want to maintain a close relationship and a close partnership with the European Union and individual member states in the future", May said at a joint press conference.
And as leaders welcomed the new tone in London and talk of a "softer Brexit" that may be less disruptive than May's clean break with the single market and customs union, officials from at least some governments saw compromise on the British bill.
Speaking as he arrived in Brussels ahead of a meeting of EU finance ministers, Hammond said the prime minister's pledges to take the United Kingdom out of the single market and the customs union remained the "broad principles" of the government's position going into the talks.
And the source told the BBC that it was understood the talks would broadly follow the EU's preferred sequence, dealing with issues of citizens' rights and a framework for calculating outstanding financial liabilities before moving on, possibly later in the year, to deal with the UK's future relationship with the EU.
Andrea Leadsom, leader of the House of Commons, said that Brexit would require a lot of legislation, including a law to enshrine current European Union rules into British law, known as the "Great Repeal Bill".
Barnier's interview will heap yet more pressure on May, who called a snap general election to give her a stronger hand at the Brexit negotiating table.
"But it is a negotiation, and as we go into that negotiation my clear view. and I believe the view of the majority of people in Britain, is that we should prioritise protecting jobs, protecting economic growth and protecting prosperity as we enter those negotiations and take them forward".
Mr Barnier, the EU's chief negotiator, has said there was no desire to punish the United Kingdom but "its accounts must be settled".
His comments come after the EU's chief Brexit negotiator, Michel Barnier, warned London to "not waste time" and begin negotiations as soon as possible.