Former British Prime Minister David Cameron has joined a chorus that is building up around Theresa May to move towards a softer stance on Brexit following the Conservative party's disastrous result in the general election.

Cameron, who had campaigned for Britain to remain in the European Union (EU) in the June 2016 referendum, had to step down as the prime minister when the result went against his stand.

His successor in Downing Street, Theresa May, had stepped in with a strong "Brexit means Brexit" message but after the June 8 general election snatched away her party's majority in Parliament, there have been growing calls for the UK to re- consider a complete breakaway from the EU.

"I think there will be pressure for a softer Brexit, Cameron said during a business conference in Poland, adding that Parliament "deserves a say" on how to manage the process.

"It's (Brexit) going to be difficult, there's no doubt about that, but perhaps an opportunity to consult more widely with the other parties on how best we can achieve it," he said.

The former prime minister, who also stepped down as an MP shortly after quitting No 10 Downing Street last year, said the 13 new Scottish Tory MPs would also help push for a softer Brexit, aided by Scottish party leader Ruth Davidson.

"There s no doubt that there is a new player on the stage. Scotland voted against Brexit. I think most of the Scottish Conservatives will want to see perhaps some changes with the policy going forward," he said.

Cameron s intervention comes soon after another former Conservative Prime Minister Sir John Major said the idea of a more robust Brexit was "increasingly unsustainable".

There are also reports suggesting that UK Chancellor Philip Hammond will seek to keep the UK in the EU customs union and with May s weakened authority in the wake of the elections, the voices against a hard Brexit within the Tory party ranks are expected to gain momentum.

Meanwhile, May has been holding talks with Northern Ireland s Democratic Unionist Party (DUP) to shore up her minority government in the House of Commons.

Any announcement of a so-called "confidence and supply deal" between the two parties is now expected to be delayed following the tragic fire in a London residential tower block early this morning.

If a deal was to be delayed it would mean the Queen's Speech, which marks the official State Opening of Parliament and had originally been planned for next Monday, could be delayed by at least a week.

It could also delay the start of Brexit talks with the EU, also scheduled to begin next week.

The Conservatives are having to rely on the support of 10 DUP MPs after they fell eight seats short of winning an overall majority of 326 at the general election.

The confidence and supply deal will ensure support for key votes in the House of Commons, such as backing the Budget and opposing any votes of no confidence.

May had said yesterday after her meeting with DUP leader Arlene Foster that the talks with the DUP had been productive and that Brexit negotiations would begin as planned next week.