Nicola Sturgeon has set out a list of demands she would make in return for backing a minority Labour government to keep Boris Johnson out of Number 10.
Ms Sturgeon made clear that Labour would need to back the "principle" of a second independence referendum.
The first minister said she would also seek greater powers for the Scottish Parliament, and an end to austerity.
But the SNP leader insisted she would not form a formal coalition with Labour if there is a hung parliament.
Instead, she said her party would potentially be willing to back a minority Labour government led by Jeremy Corbyn on an issue-by-issue basis.
The Conservatives said Ms Sturgeon's comments meant the UK could be facing referendums on both the EU and independence next year if Labour wins the election.
And Labour said it was aiming to win the election and was therefore "not in the business of talking about deals with other parties".
The SNP won 35 seats in the last general election in 2017, making it the third largest party in the House of Commons and leading to speculation that it could hold the balance of power if no one wins an outright majority in the election on 12 December.
But Ms Sturgeon again warned that other parties "need not bother picking up the phone" to her after the election unless they are willing to support her party's right to hold an independence referendum next year.
She said her party would drive a "hard bargain" with anyone seeking SNP support in the event of a hung parliament - and explicitly ruled out doing any deal with the Conservatives.
Ms Sturgeon's list of demands for SNP support included the "principle of the people of Scotland deciding our own future", as well as the devolution of powers over immigration, employment and drug laws.
Prime Minister Boris Johnson said during a visit to Scotland on Thursday that he would fight to "keep our fantastic United Kingdom together and prevent another referendum next year".
The Liberal Democrats are also firmly opposed to another independence referendum - while Labour leader Jeremy Corbyn has said indyref2 is neither "desirable or necessary" and has indicated he would not support one in the "formative years" of a Labour government.
Speaking to the BBC's Laura Kuenssberg after the launch of her party's election campaign, Ms Sturgeon said that the SNP holding the balance of power after the election would put Scotland in an "incredibly influential position".
She stressed that she would "never put Boris Johnson in power" and claimed that people had "good reason" to be concerned about the prospect of Mr Corbyn becoming prime minister.
But she insisted it would be better to have SNP MPs "in there making sure the right issues are progressed and the right values upheld".
The SNP leader had earlier said the vote on 12 December would be the "most important in our lifetime", and said the outcome would determine the future of the country for "generations to come".
Ms Sturgeon said Scotland had been on a "journey" in the two decades since the creation of the devolved Scottish Parliament.
But she claimed that "many of the gains of the last 20 years and the promise of a better future" were now under threat from what she described as "hardline Brexit ultras" within the Conservative Party.
Ms Sturgeon said: "Scotland's vote to remain in the EU has been ignored.
"The Conservative Party has ridden roughshod over the Scottish Parliament. For the first time ever the UK government has chosen to legislate on devolved matters without the consent of Holyrood.
"With so-called 'moderate' Conservatives in full retreat and the hard-line Brexit ultras on the march, that is surely only a taste of what is to come."
Ms Sturgeon said a vote for the SNP was therefore a vote to "escape Brexit" and to "take Scotland's future out of the hands of Boris Johnson and a broken Westminster system".
And she said the SNP already had "a cast iron mandate" for another independence referendum and warned the prime minister that he has "no right to block the democratic wishes of the people of Scotland".
Ms Sturgeon also unveiled plans for the SNP to launch a bill at Westminster aimed at protecting the NHS across the UK from privatisation and future trade deals.
The NHS Protection Bill would block any UK government from using the NHS as a "bargaining chip" in trade talks.
If passed it would also give devolved parliaments in Scotland, Wales and Northern Ireland a veto on any deal, she said.
Ms Sturgeon claims that despite health policy being devolved to the Scottish Parliament, the UK government could still "sell off" the NHS in trade negotiations.
The UK government has insisted the NHS is "not on the table" for trade talks and is not in any way "up for sale", pledging to bring forward the "biggest programme of NHS investment in a generation" if the Conservatives win the election.
Nicola Sturgeon does not rule out a deal with Mr Corbyn, or Labour, in the event of a hung Parliament. She has a shopping list which seems to grow by the day - from anti-austerity economics through the enhanced devolution of benefits through action on climate change.
But top of that list is a further referendum on independence. Might Mr Corbyn be persuaded to adopt that? He won't say yes or no. It isn't a priority, we are told. No deal, no pact. Labour would govern as a minority and challenge others to cast their votes on individual issues.
But then successive members of Team Corbyn cloud the issue once more by hinting it might arise a bit down the road.
Would Ms Sturgeon be willing to wait? Perhaps - although she insisted at the launch that the timetable should be determined by the Scottish Parliament, not at Westminster.
This potential accommodation, of course, does not operate in a vacuum. Particularly in Scotland, each of the other parties is advancing a case with regard to these twin issues of Brexit and independence.
What has the reaction been?
Michael Gove of the Conservatives said Ms Sturgeon had dropped a "bombshell" onto the election campaign by again saying she would seek a fresh independence referendum as a price for supporting Mr Corbyn in government.
Mr Gove said: "That would mean there would be two referendums next year - one on Europe, and one on Scotland's independence. It's the last thing this country needs.
"We need to get Brexit done and get on with the people's priorities, but the SNP and Labour instead want more misery as two referendums consume all the air in our political system."
The Shadow Brexit Secretary, Keir Starmer, said Labour would not be doing deals with anyone when asked about the party's position on a second independence referendum in Scotland.
Speaking on the campaign trail in Wales, Mr Starmer said: "The Labour party is in this election for real change and we're in it to win it, and therefore we're not in the business of talking about deals with other parties."
Lib Dem leader Jo Swinson said her party would be willing to work with other parties at Westminster, but cautioned: "When it comes to the SNP, they need to drop their obsession with independence because we've just had three years of Brexit chaos that's hurting our country, that's hurting our public services."
"We need to learn the lessons of Brexit not repeat the mistakes."
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