The Tories lost 13 seats at the General Election, and despite May's hopes of securing a strong majority, the UK has been left facing a hung parliament.
The Prime Minister is still in talks with the Democratic Unionist Party over a potential "confidence and supply" deal to allow the Conservatives to press on, but Gove has said the Tories must also show flexibility on Brexit.
Speaking on the Today programme, Gove - a leading figure in the Vote Leave campaign - said that it was time for an "open conversation" about how the UK leaves the EU.
"It is also the case that we need to recognise that we, as Conservatives, were not returned as a majority. That means we need to proceed with the maximum possible consensus," he said.
"We also need to ensure that the concerns of people who voted Remain, many of whom want us to press ahead with leaving the EU as quickly and in as orderly a fashion as possible, we need to make sure their concerns are part of our conversation."
It comes as former Tory leader Lord Hague has written in the Telegraph calling for a cross-party alliance to help steer Brexit, with leading business groups also handed a role.
"Call in the CBI, the Institute of Directors, the British Chambers of Commerce, the Federation of Small Businesses, the TUC, the first minsiters of the devolved administrations and the leaders of all the opposition parties - yes, even Corbyn - leading MPs of all parties and say: 'If you are willing to discuss how to make this work within these parameters, come in and we will be open to your views,'" Hague said.
Scottish Tories leader Ruth Davidson similarly called for Brexit plans to be "re-opened" in the aftermath of last week's results, while former small business minister and repeated rebel Anna Soubry has called on British firms to "speak truth to power" on Brexit.
But pressed on whether the government's plans would change, Brexit secretary David Davis yesterday reiterated the commitment to quitting both the Single Market and the European Customs Union.