First Minister Nicola Sturgeon has said there are "significant questions" about Alex Salmond's political comeback.
Her comments came after Mr Salmond revealed he would be among the new Alba Party's regional list candidates.
The former first minister said his aim was to build "a supermajority for independence" after the May election.
In response, the Scottish Conservatives wrote to Scottish Labour and the Scottish Liberal Democrats and urged them to work together.
But Scottish Labour leader Anas Sarwar accused the party of playing games and added: "This election cannot be about an SNP psychodrama."
And Scottish Liberal Democrat campaign chairman Alistair Carmichael insisted that the politics of Tory leader Douglas Ross were "far too dark and divisive".
As the news of the new party broke on Friday, other Holyrood leaders questioned Mr Salmond's character.
Speaking on the campaign trail on Saturday, Ms Sturgeon said: "I take no pleasure whatsoever in saying this but I think there are significant questions about the appropriateness of his return to public office given the concerns that have been raised about his behaviour previously but that's for voters to judge and decide.
"This is an election. We live in a democracy. For my part, in this campaign, I am focused on the interests of the country."
She added that the Covid pandemic and its consequences made this "probably the most serious election in any of our lifetimes".
Asked about Mr Salmond's vision of securing a "supermajority", Ms Sturgeon said: "I know Alex Salmond very well. He makes big claims which often don't stand up to scrutiny."
The first minister added her predecessor has changed his mind on how to secure independence due to "self interest and, dare I say it, ego".
She added: "Alex Salmond is a gambler. It is what he enjoys doing. But this is not the time to gamble with the future of the country."
Meanwhile, on the first weekend of the campaign it emerged that MP Kenny MacAskill has left the SNP to join the Alba Party.
The former justice secretary will stand as a candidate on the Lothian regional list. He is now the Alba Party's first MP at Westminster.
During an online launch on Friday Mr Salmond said the party planned to field at least four candidates on the regional lists in every part of the country.
People have two votes in Holyrood elections - one for a constituency MSP, and another in a regional ballot designed to make the overall result more proportional.
Mr Salmond claimed that if Alba won regional list seats, this could lead to there being 90 or more MSPs at Holyrood who support independence.
He added that Prime Minister Boris Johnson, who has already dismissed the SNP's calls for a second independence referendum, would "find it much more difficult to say no to a parliament and a country."
SCOTLAND'S ELECTION: THE BASICS
What elections are happening? On 6 May, people across Scotland will vote to elect 129 Members of the Scottish Parliament (MSPs). The party that wins the most seats will form the Scottish government. Find out more here.
What powers does the Scottish Parliament have? MSPs pass laws on most aspects of day-to-day life in Scotland, such as health, education and transport. They also have control over some taxes and welfare benefits. Defence, foreign policy and immigration are decided by the UK Parliament.
How do I vote? Anyone who lives in Scotland and is registered to vote is eligible, so long as they are aged 16 or over on the day of the election. You can register to vote online.
In response to the new party, Scottish Conservative leader Douglas Ross requested a meeting with Mr Sarwar and Scottish Liberal Democrat leader Willie Rennie in a bid to "rediscover the Better Together spirit".
Mr Ross asked his counterparts to sign a Unionist manifesto which pledges to vote against a second independence referendum regardless of the result in May's election; rule out a coalition or confidence deal with any party who seeks to hold an independence referendum in the next parliament; and agree to form a pro-UK, anti-referendum coalition if the opportunity arises.
He also warned that another independence poll would be divisive.
Mr Ross said: "The prospect of that referendum, held without the UK government's consent, has increased dramatically with the formation of Alex Salmond's party. This new Nationalist party is designed solely to deliver an independence super-majority.
"The threat of a Scottish Parliament focused entirely on breaking up the UK for the next five years, instead of on the essential task of rebuilding our country, is now very real."
As the biggest opposition party, Mr Ross vowed to stand up to the SNP but added: "If we can work with other pro-UK parties to further our chances of preventing a nationalist government, we will do so."
Mr Sarwar responded to the letter by confirming Scottish Labour did not support Scottish independence or a second referendum.
But he also said the focus of the election must be how the country emerged from the pandemic.
Mr Sarwar said: "It cannot be about Nicola Sturgeon and Alex Salmond settling old scores. And it cannot be about your petty agenda of game playing.
"This election must be about the people of Scotland, their families and a national recovery. As you have recognised yourself, one of the largest threats to the Union is the leader of your own party."
The new Scottish Labour leader said he did not want to waste time and energy "on the imagined conflicts that divide us" and would rather focus on major issues, such as child poverty.
He concluded: "Rather than entertain your latest desperate plea for attention, I am focusing our energy on what matters - guaranteeing a fairer recovery and a stronger Scotland.
"Scotland deserves a better government and - as your letter demonstrates - it deserves a better opposition."
Scottish Liberal Democrat campaign chairman Alistair Carmichael said: "Lib Dems will work with others to deliver a constructive and ambitious plan for recovery, but Douglas Ross's politics are far too dark and divisive.
"We will focus on winning seats and ensuring that the next government is focused on putting the recovery first, not independence.
"As a football referee Douglas Ross has a knack for uniting the fans of opposing teams. As a party leader he seems to do the exact opposite."
On Saturday, Patrick Harvie, co-leader of the Scottish Greens, said: "I think a great many people in Scotland want an election based on the big issues facing our country. Not just the constitutional future but the response to the pandemic, how we build a green recovery from the pandemic that creates a fairer and more equal society, the climate crisis - especially with COP26 coming to Glasgow - and the poverty and inequality that still blights our society.
"I don't think Alex Salmond is the answer to any of these questions. He's clearly a discredited figure in Scottish politics."
Mr Harvie said the behaviour that Alex Salmond had admitted - ignoring the criminal offences he was acquitted over - would have seen summary dismissal for gross misconduct from many other jobs.
"I don't think any credible party would have him as a candidate today," he added.
In 2019, the Scottish government admitted it had acted unlawfully while investigating allegations of sexual harassment against Mr Salmond, which he strongly denied.
The following year he was acquitted of all 13 charges of sexual assault after a trial at the High Court.
Giving evidence, Mr Salmond said that in hindsight he wished he had been "more careful with people's personal space".
His defence lawyer, Gordon Jackson QC, told the court Mr Salmond had not always behaved well and could have been "a better man on occasions" - but had never sexually assaulted anyone.
The Alba's Party's unexpected launch came at the end of an dramatic week at Holyrood.
On Wednesday, Mr Salmond said he would take fresh legal action over the conduct of the Scottish government's top civil servant.
A report by MSPs on Tuesday described the government's handling of harassment complaints against Mr Salmond as "seriously flawed".
The previous day, First Minister Nicola Sturgeon was cleared of breaching the ministerial code over her involvement in the Alex Salmond saga.
- LAST DAYS BEFORE LOCKDOWN: 5 live asks listeners how they spent their last few days before the UK's first lockdown
- MENTAL HEALTH IN COVID: Giving you the tools to look after your mental health during the pandemic