Neil Hamilton is expected to challenge Nathan Gill to lead UKIP's new assembly group, sources have told BBC Wales.
The former Tory MP has denied a suggestion by a senior UKIP source that he will try to change party rules giving UKIP leader Nigel Farage sole power to appoint the leader in Wales.
Current UKIP rules allow different politicians to lead the assembly group and the party in Wales.
A source close to Mr Gill said he was confident most UKIP AMs would back him.
Mr Hamilton and Mr Gill were among seven new UKIP AMs elected on Thursday.
Gareth Bennett, Mark Reckless, David Rowlands and Mr Gill took the oath of office at the Senedd in Cardiff Bay on Saturday.
Mr Hamilton and another newly-elected AM - Caroline Jones - were present but were not sworn in.
Party sources expect Mr Hamilton to make the leadership challenge on Tuesday to Mr Gill, who was appointed UKIP Wales leader by Mr Farage in 2014.
Mr Hamilton and Ms Jones are expected to take the oath on Tuesday morning, after attending a meeting of UKIP's ruling National Executive Committee (NEC) on Monday.
At the meeting they will stand down from the NEC, which is required under UKIP's rules now that they have become elected politicians.
A senior UKIP source suggested Mr Hamilton might use the meeting to attempt to change party rules stating that UKIP's national leader has the sole power to appoint the party's leader in Wales.
Mr Hamilton has denied he has such a plan.
Earlier on Saturday, Mr Hamilton told BBC Radio Wales that UKIP AMs would agree next week who should lead their group.
He stressed that Mr Gill was not automatically in charge and said there were "many people in UKIP" with the right skills for the the job.
"We're going to have a collective discussion when we all meet together and we'll come up with a decision as to which of us is likely to be the most effective leader in the assembly," he said.
Asked if he would be the best candidate, Mr Hamilton added: "I, of course, have a lot of parliamentary experience and so has [fellow UKIP AM and ex-Tory MP] Mark Reckless.
"I've been around in politics for a very long time at quite a high level, and we have many people in UKIP who've got the kind of skills which will be advantageous to a leader.
"But I'm not going to pre-empt the discussions that we will have, probably on Tuesday."
First Minister Carwyn Jones is spending the weekend preparing to form a new government, after Labour won 29 of the 60 seats, losing just one to Plaid.
The new assembly is expected to hold its first meeting next week.
Discussions are ongoing between the parties to decide who should be presiding officer and deputy presiding officer, and take charge of assembly proceedings.
The Conservative assembly group will meet on Monday to discuss strategy, after a disappointing election under Welsh leader Andrew RT Davies.
The party failed to win any target seats and lost three to fall into third place behind Plaid Cymru.
Monmouth AM Nick Ramsay said the Conservatives' problems could not be "brushed under the table" and the leadership needed to be discussed.
He called it a "difficult" campaign but backed Mr Davies.
"The 11 of us that are now left need to look at where we go from here," he said.
"[We need to] decide how we get there and then look at all issues, including the leadership, but other issues as well such as the strategy the group adopts."
A Welsh Conservative spokesman described the campaign as "very positive" and "ambitious" with "great ideas".
"There's a fifth party in play now, which has affected everyone else," he said, referring to UKIP.
"We've proven we can challenge in key seats next time around."
Plaid Cymru AM Rhun ap Iorwerth said his party would be "constructive but playing the crucial role of a strong opposition" to Labour.
"We will use that wisely to put forward a vision for the country which is as clear now as it was the day before the election," he said.
Kirsty Williams, the only surviving Lib Dem AM, has resigned as leader of the party in Wales after it lost four of its five Senedd seats.
Analysis by Daniel Davies, BBC Wales political correspondent
Labour looks likely to form another minority Labour administration after an election in which only one constituency changed hands.
And yet, the election has revealed a big change in Welsh politics - a change that has arrived in Cardiff Bay in the form of seven UKIP AMs who won seats on the regional lists.
They say they want to shake the place up. But Nigel Farage also says they want to play a constructive role.
I'm not sure how you achieve both.
Either way, there's a question facing UKIP - who will lead them in Wales?
Nathan Gill was appointed UKIP Wales leader by Nigel Farage in 2014 - a job he's compared to "herding cats".
It sounds as though Neil Hamilton - the former Tory MP whose political career has been re-booted by this election - would be only too happy to relieve him of his duty.
But his elevation is unlikely to heal rifts in the party. Even the suggestion that he could be a candidate in this election caused a row in UKIP's ranks.