Mr Swinney’s comments come as a new Prime Minister is expected to be in place by the end of next week, following Liz Truss’s resignation last week after only 44 chaotic days in office.
He said the Tories must realise they are “no longer fit to govern”.
Mr Swinney said: “This is the moment after all the failures of the Conservative Party, its divisions and its lack of leadership, to give the people of this country a chance to elect a new Parliament and decide how it should be governed.
“I think we should have a general election and I don’t think the Conservative Party should be allowed to play any more of their games any longer.”
The Deputy First Minister added: “The people of this country must be given the chance to decide on the way we are governed moving forward.
"I think they should do the decent thing and allow the people of this country to have their choice. If we put enough pressure on them the Conservative Party has to realise the mistakes they have made and be held to account for those mistakes.”
However, Craig Hoy, the Scottish Conservatives chairman, said the party needed to “provide stability” by having a “very quick” leadership election, instead of a two-month long general election.
Mr Hoy said: “A general election would only provide eight weeks of huge political instability with all political parties taking their eye off the ball at a time where we should be focused on making sure we help households in every part of the UK through economic challenges that lie ahead.”
Asked if he thought the UK population should have a say in the next PM, Mr Hoy said: “We have seen huge economic instability, we have a global cost of living crisis, the war in Ukraine and we are dealing with the effects of the Covid crisis. We have been through a lot of political instability and I think my party has to take some of the blame for that but I think what we now need to do is create the conditions for economic and and political stability and that’s what a new Prime Minister will do."
Mr Hoy compared Tory members re-electing a new PM, to Nicola Sturgeon taking over from Alex Salmond as First Minister stating, “She [Ms Sturgeon] didn’t call for a Scottish parliamentary election.”
On the leadership contest, Mr Hoy said: “This is well within the constitution and the rules and I think what the Scottish people and British people now want us to do is to govern in their interests which is to make sure we seriously tackle the issues we see in the economy.”
Yet, prominent Scottish Conservatives are keeping their support for potential candidates in the leadership race hidden.
Both Douglas Ross and Alister Jack will not be revealing who they are supporting as the next PM, according to sources.
A Scottish Conservative spokesperson told The Scotsman the Scottish Conservative leader would not be voicing his personal choice in the PM leadership.
Mr Ross recently told the press he made his views on Boris Johnson “well known” as he previously said having Johnson as PM would be a “disaster”.
A spokesperson from the Scotland Office said: “Alister [Jack] has not come out in support of any candidate, and didn’t the last time.”
Asked who he thinks should be the next UK Prime Minister, Mr Hoy said, as chairman, he would remain impartial throughout the process.
A handful of Scottish Conservatives MPs have voiced their support for declared candidates, including Miles Briggs and Andrew Bowie, who have come out in favour of Rishi Sunak, while John Lamont has backed Penny Mordaunt for Prime Minister.
Backing Mr Sunak as PM, Mr Bowie, MP for West Aberdeenshire and Kincardine, said he was “proud to support him” and, as the country faces “profound challenges”, he added “Rishi Sunak is the person to tackle them”. Meanwhile, Mr Lamont, MP for the Borders, said: “Penny has the values, integrity and vision to unite our country and our party.”
Asked to comment on whether the Scottish Conservatives could work with Mr Johnson if he were to become PM for a second time, Mr Hoy said: "We don't even know if he will throw his hat into the ring".
“On Monday, we are going to find out one, two or potentially three candidates and then we will have an indicative vote of conservative MPs at Westminster that will show where the views of the parliamentary party are and that will go to the country.”
However, the final vote for the next PM is expected to go to the thousands of Tory party members across the country.
The first voting phase usually sees the number of candidates whittled down, with a series of ballots voted upon by Tory MPs and the next phase tends to see the two MPs remaining face a vote of all Conservative Party members, with the winner becoming the next Tory leader and Prime Minister.