"I think there will be pressure for a softer Brexit", Mr Cameron added, saying that Parliament now "deserves a say" on the issue.
For the officials sitting down on Monday, at least on the European Union side, a major worry is Britain crashing out into a limbo, with no deal.
"If the government cannot even secure a deal with the DUP, how on earth can they get a deal with the European Union?" asked MP Alistair Carmichael.
Britain's finance minister said Friday that jobs and growth should be the priority when Brexit negotiations begin next week, indicating that the weakened government in London might be softening its tone with Brussels.
Mr Davis will meet the European Commission's chief negotiator Michel Barnier from Monday before being joined by Theresa May on Thursday who will deliver a speech to EU leaders on what the election means for Brexit. He's been deputy to the Brexit Secretary David Davis and involved in preparation for the negotiations.
Following more than an hour of talks between May and DUP leader Arlene Foster on Tuesday, May said the discussion had been productive and Foster said she hoped a deal could be reached "sooner rather than later".
"What we are doing in relation to the productive talks that we are holding with the Democratic Unionist Party is ensuring that it is possible to, with their support, give the stability to the UK Government that I think is necessary at this time".
"The fact they are coming and that they agree to talk about the subjects that we set out, shows that the clash is under control", a senior European official told AFP on condition of anonymity.
The British government wants the negotiations to include the future relationship with Europe and an all-important trade deal with the bloc.
Ordinary Britons are also beginning to feel the cost of Brexit because of higher import prices caused by a plunge in the pound and businesses are increasingly anxious about losing trade access.
Mr Hammond said the United Kingdom would definitely be leaving both the EU single market and the customs union, but said that a transitional deal to enable a smooth exit from both was highly desierable.
May reportedly faces "civil war" inside her Cabinet, over whether she should continue with her hardline position on Brexit.
The Times also reported that party members who had campaigned to keep Britain in the European Union were likely to have a candidate lined up to replace May, with interior minister Amber Rudd the likely option.
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