Nicola Sturgeon has said she "emphatically" wants to remain as first minister for at least a few years.
Appearing on The Andrew Marr Show, the SNP leader said she believed she had the support of "party and of country" to hold her post.
It comes after party figures told BBC political correspondent Nick Eardley Ms Sturgeon may be in trouble.
She also defended a proposal for a "Scottish visa" system after the UK government unveiled immigration plans.
MSPs would decide the criteria for this new visa, and the Scottish government would receive and assess applications before sending them to the UK government for security checks.
The UK's post-Brexit system, which was announced earlier this week, means that low-skilled workers would not get visas.
However Ms Sturgeon's position is that this would cause "devastation" for Scotland's economy as it would reduce the number of people entering the country with "restrictive" border controls.
There are concerns about Scotland's ageing population and shrinking workforce, with the National Records of Scotland projecting that deaths could outweigh births over the next 25 years.
Scottish ministers say this means greater inward migration is needed to boost Scotland's working-age population in particular, and that an end to freedom of movement could threaten this.
In a letter to Number 10 published on Sunday, Ms Sturgeon has called for a meeting with Boris Johnson to discuss immigration policy.
She told Andrew Marr she hopes to take a delegation of sectoral and business leaders to Downing Street to set out arguments for a different Scottish system.
She said: "You have a UK government that has as an expressed objective - reducing the number of people who come into the UK from other countries.
"My point is that that objective - in and of itself - is deeply damaging to Scotland's economy and our future prosperity.
"It will make us poorer and that is why I really want to see this change and for Scotland's interests to be recognised."
The UK government, however, has urged employers to "move away" from relying on "cheap labour" from Europe and invest in retaining staff and developing automation technology.
And the Migration Advisory Committee has said Scotland's needs are "not sufficiently different" from the rest of the UK to justify a "very different" system, with the north of England facing similar issues.
Ms Sturgeon said there was "not a shred of evidence" to support the idea that Scottish jobs had been undercut and called for more powers over employment law to ensure fair working conditions.
She said: "There is much evidence to the contrary including the views of the migration advisory committee - that immigration and EU immigration in particular does not drive down wages, either in the Scottish economy or in the UK economy.
"Migrants make a net contribution to our economy. If we have a problem of low wages or working conditions not being what they should be, that's about poor regulation in the UK economy."
A UK government spokesman said: "Our new points-based immigration system will work in the interests of the whole of the United Kingdom.
"We will continue to work with stakeholders and industry in Scotland to ensure the new proposals work for all sectors."
Future as leader
Earlier this week Nick Eardley wrote that a number of SNP figures said Ms Sturgeon may have to "fall on her sword" amid increasing discussions over her future.
When asked if she would remain in her position as leader of Scotland, Ms Sturgeon said there were two conditions to consider.
She said: "Firstly you have to have the support, not just of party but of country, and I would say humbly that I've just led my party to another landslide election victory.
"Secondly I have to be sure that I want to do this job, think I'm the best person to to this job, have the drive and energy - and that is emphatically the case."
Marr also pushed Ms Sturgeon on whether she would look to hold another referendum on Scottish independence this year, despite the prime minister's flat refusal.
She reiterated her call for independence supporters to "be patient" but said it was important to continue arguing for another vote as the UK government negotiates its "future relationship with Europe".
"I think it's important that Scotland decides whether or not it wants to go down that road and if it doesn't we start to plot a better route forward," she said.
Ms Sturgeon added that she does not rule out "testing the limit of the power of the Scottish Parliament" in court - but it was not an option she was "actively looking at".
She has previously ruled out the possibility of holding an unofficial referendum similar to the one in Catalonia in 2017.