Theresa May's Cabinet and party colleagues have broadly given her backing during a series of TV interviews.
For now, both men, and the rest of the Conservative party, appear to be backing May, despite media speculation that rivals were lining up to take her place.
Asked if that meant no transitional arrangements, she replied: "Well, I'm extremely optimistic that we will find there is a lot we can agree on".
Theresa May had called for voters in the election to choose her "strong and stable leadership" to ensure that the United Kingdom had a strong hand in the Brexit negotiations.
The election debacle has left her weakened and battling for her political survival. 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.
With Mrs May still hammering out the details of a post-election deal to stay in power with the support of a small Northern Irish party, there are fears of a disorderly exit that would weaken the West, imperil Britain's US$2.5 trillion (S$3.5 trillion) economy and undermine London's position as the only financial centre to rival NY.
However, the speech was cancelled following the Grenfell Tower tragedy in West London. However, the shock result and loss of a Conservative majority has reopened the debate over what sort of Brexit Britain wants.
Ruth Davidson, an enthusiastic Remainer and now a very influential figure in the Tory party after helping her Scottish Conservatives increase from only one seat in the Westminster Parliament to 13, told May that she must now pursue a softer "open Brexit" that prioritizes economic prosperity over immigration control.
Davis, a prominent "Leave" campaigner in the referendum, said he was approaching the talks in a "constructive way", knowing they will be "difficult at points".
Monday's negotiations are to open at 11:00am in Brussels (0900 GMT) with 90 minutes of talks between Barnier and Davis, followed by a working lunch between the pair and a press conference.
The Frenchman has been given a mandate by the leaders of the remaining EU27 states to focus initially on the key elements of the withdrawal package - a multi-billion euro "divorce bill", rights for European Union citizens and the status of the border between Northern Ireland and the Republic.
Once sufficient progress has been made on the withdrawal issues, the European Union say they will open discussions on a new trade agreement.
The government on Saturday said parliament would hold a special two-year session starting this week, sitting for double the normal time to allow it to overhaul European Union legislation.
She said: "There is not the parliamentary arithmetic that is going to back a hard Brexit however much some colleagues in both parties would like to see that happen ... we are not going to end up, I don't think, with a hard Brexit".
That the British were ready to break negotiations down on the lines of priorities identified by Brussels implied, he said, a willingness to start a first phase of divorce talks before moving on next year to a second phase on the new free trade relationship May wants - a "sequencing" London has disputed.
Britain's Brexit ministry said the team travelling to Brussels was confident it could achieve a "bold and ambitious deal" and forge a new, close arrangement with the bloc.
- US Open snippets from fourth round at Erin Hills
- Reports swirl about Beyonce's twins, but no confirmation yet
- US Open results: Brooks Koepka wins golf championship with record-equalling score
- Fakhar, Amir sparkle as Pakistan stun India in Champions Trophy final
- US Senate Overwhelmingly Backs Stronger Sanctions Against Russia, Iran
- Murray sweeps past Khachanov into French Open quarters
- Trump lawyer: US President not under investigation in Russia probe
- Koepka wins US Open Title
- US Open sadness for Rickie Fowler, Justin Thomas
- Trump Confirms He Is Under Investigation Over Comey Firing