Muller told the New Zealand Herald he expected to set out the reshuffle of his caucus and his office today, and said that how people voted in the leadership election would not be a factor in his decision making.
He also warned that National MPs did not have time "to brood" with an election four months away.
The Herald has learned that Bridges was originally offered the justice portfolio, but has rejected it and instead asked for foreign affairs.
Gerry Brownlee is expected to keep that portfolio, and Muller turned Bridges down. It remains unclear what Bridges will now get – if anything.
Muller confirmed he had spoken to Bridges, but would not say what the two talked about beyond saying Bridges had been "remarkably professional".
Bridges would not comment beyond saying he was yet to make decisions about his future.
Brownlee is expected to also take the election campaign chair role from former deputy leader Paula Bennett.
The only positions announced so far are Paul Goldsmith in finance, while Muller himself will take the small business portfolio.
As well as Goldsmith, another Bridges' supporter, Mark Mitchell, is likely to be on the front bench. Others tipped for the front bench include Muller supporters deputy Nikki Kaye, Shane Reti, Nicola Willis, Judith Collins and Chris Bishop.
Neither Bridges nor Bennett will get front-bench positions and Bennett's fate remains uncertain.
The reshuffle will be critical if Muller is to restore unity in his caucus after a bruising leadership showdown. In the past, new leaders have given front bench spots to those who opposed them to try to heal the wounds - Bridges put Kaye, Amy Adams and Judith Collins on his front bench in 2018.
Muller said change was inevitable after such a leadership change but he intended to make his appointments on merit.
"I can't sit here and say the last few days have been all sweetness and light, it never is. Of course it was tough.
"Transitions of leadership in that manner are always tough. But unlike other times in political cycles when you have two or three years to brood, we've got 120 days to go.
"How people vote, or signalled who they had preference for will not be at all in the front of mind when I bring the best group of people together."
He said he did believe MPs knew the change was working.
Muller's first weekend in the role was also marked by a debate about a Make America Great Again cap in his office – part of his collection of political paraphernalia.
After Muslim groups criticised its presence, Muller said it did not reflect any admiration of Trump or his views.
"I've been asked directly do I support President Trump's style, and I say 'not at all'.
"I find his style of politics appalling.
"Hopefully they can understand I am a respectful person, and that my collection of political paraphernalia doesn't in any way seek to align myself to any particular political positioning of any individuals offshore."
Asked if it was wise to publicly describe Trump as "appalling" given Muller might one day be prime minister while Trump was President, Muller said he did not think Trump would be bothered.
"My experience of observing the President of the United States is he has a remarkable comfort in his own world view, and his own perspective of other people. I don't think for one moment that he would, in any way, be personally slighted or frustrated if a New Zealand prime minister held a different view to his approach to the world.
"There is no problem in being able to hold a view on these matters, and I have every confidence that the President would not be fazed at all."
Muller is also expected to set out his new staff and campaign teams early this week.
The National Party board, rather than the leader, appoint the campaign positions. However, the leader is on the board and his view carries weight.
A list leaked to the NZ Herald also lists Amy Adams in a policy position and Adams has also been tipped to get a Covid-19-related portfolio.
Adams has said she was retiring from politics at 2020, but there are reports she has now changed her mind and will stand on National's list – taking the list-only slot that was reserved for Bennett. Adams is yet to confirm or deny those reports.
A former Beehive staffer, Tim Hurdle, who is now on Muller's Bay of Plenty electorate branch, was listed as campaign director.
Bridge's chief of staff Jamie Grey and chief press secretary Rachel Morton both resigned on Friday.
