The EU has said that sufficient progress must be made on these issues before trade deals can be discussed, though Britain had argued the talks should take place simultaneously.
Mr Gove's appointment came after former chancellor George Osborne branded her a "dead woman walking", warning that she could be ousted from Downing Street in a matter of days.
If the announcement of the deal is delayed, then the Queen's speech, which is supposed to happen on Monday, June 19, could also be delayed by at least a week.
The Labour leader received a standing ovation from his MPs before telling Mrs May: "Democracy is a wondrous thing, and can throw up some very unexpected results". The House of Commons will also choose a new speaker and Lords will be sworn-in.
The British prime minister meant to secure her Brexit mandate when she announced the elections on the 18th of April, because "The country is coming together but Westminster is not".
Meanwhile Denmark's finance minister said he hoped the outcome of the British election would lead to a pause to rethink the direction in which the Brexit negotiations were heading."After the general election it is very unclear to know exactly what the United Kingdom approach to Brexit now will be", Kristian Jensen said.German Foreign Minister Wolfgang Schaeuble, speaking at a financial conference in Berlin, said his government was aiming for a Brexit deal that would limit negative consequences for the bloc but nor did it want to weaken Britain.
The Conservatives are considering an arrangement in which the Northern Ireland party backs May on the budget and her confidence motions.
Despite her party's expectations of a landslide victory Ms May lost her majority in parliament, pushing her into rushed talks on a support agreement with a small Eurosceptic Northern Irish Protestant party with 10 parliamentary seats.
While peace in Northern Ireland was not in danger of collapsing imminently, it could "unwind".
Meanwhile, sources from Downing Street told the channel that talks of a delay in the announcement of the deal were "not coming from us".
"She acknowledged that it was a mistake to drop that into the middle of an election campaign", one MP said, adding that they would be "amazed" if the policy was not dropped from the upcoming Queen's Speech.
"I got us into this mess so I'm going to get us out of it".
But there was potential discomfort for the government when Ian Paisley Jr, one of the DUP's ten MPs and the son of its firebrand founder, underlined its links with organised Unionism and the Orange Order.
As Mrs Foster met with her MPs in Westminster, she said: "The future's bright", prompting Ian Paisley junior to respond: "The future's orange".
May earlier reappointed most of her ministers to cabinet but brought a Brexit campaigner and party rival into government to try to unite her Conservatives.
In contrast to May's disappointing election result, Macron's party is expected to claim between 415 and 445 seats in the lower house when next Sunday's snd round of voting concludes.
"We've just been returned to government with a minority government in effect; it's our duty to make it work, it's our duty to make it deliver for the British people".
"I hope that we can reach a conclusion sooner than later".
Pulling together a government is proving to be quite the headache for Theresa May.
Labour said the government was "in chaos" and continued to be "in denial" about the message voters had sent about their opposition to an "extreme Brexit".