The Liberal Democrats say they "intend to lead Wiltshire moving forward", despite not having a majority.
For the first time, the Lib Dems became the biggest party in last week's local elections, but fell short of the 50 seats needed to control the chamber.
The Conservatives say it is premature for the Liberal Democrats to already be forming their cabinet as there is no guarantee of how the new administration will function.
The key Wiltshire Council meeting to appoint a new leader, where any power sharing deals are first tested, takes place on Tuesday 20 May.
The Liberal Democrats in Wiltshire even had a visit from national leader Sir Ed Davey at the weekend after the result – he played cricket with the county's Lib Dem leader, Ian Thorn.
"We won the popular vote. We think it's absolutely right that the Liberal Democrats should lead any administration in Wiltshire going forward," said Mr Thorn.
He is very insistent his party will lead the council and said he does not see discussing a cabinet as premature, but instead wants to show that the party is "determined" and make it clear there is a "first class team to take us forward".
That would appear to include inviting an independent councillor Jon Hubbard to join their ruling cabinet, Mr Thorn suggested on social media.
Analysis: Dan O'Brien, Wiltshire political reporter
It's an obvious fact that suddenly becomes important – if you want to get something done in a council chamber, you need to persuade the majority of councillors to vote in your favour.
At County Hall, that means persuading at least 50 of the 98 councillors to pick your side. The Liberal Democrats, now the largest party, have 43.
They strongly believe that with the most councillors, and winning the popular vote, they now deserve to run the show – but there are no guarantees for how this will all play out.
They're promising to listen to all sides, but it is hard to envisage the Conservatives or Reform UK's councillors backing too many Lib Dem ideas.
So the way things are heading, they'll push to persuade the independent group and the one remaining Labour councillor to team up – only just getting them over the line.
Whether we see a formal coalition or something less official, we don't yet know – but the Lib Dems have already published their hopes for an independent councillor to take a seat on the cabinet.
It's possible this leads to a more collaborative form of politics and decision making – that's what the Lib Dem leader is hoping for and there are plenty of councils across the land who make this sort of setup work.
But the approach is yet to be tested against Wiltshire's new political reality.
