Mr Yousaf said he would launch a consultation with the rail unions and political parties.
Scottish ministers have been given the power for such a bid from the UK Government, which they did not have when the current franchise was awarded to Dutch firm Abellio in 2014.
Options could include a model similar to CalMac, the Scottish Government-owned company which won a new contract to run west coast ferries in May after beating off private sector rival bidder Serco.
A public sector bid for ScotRail was in the SNP’s Scottish election manifesto this year.
The current ten-year franchise is due to end in 2025, but Mr Yousaf said today it could be terminated early if ScotRail’s punctuality falls 5.5 percentage points from the current level for three months.
It is currently 89.8 per cent - 0.5 points below the acceptable threshold.
Ministers could also exercise an option in 2020 to end the franchise in 2022.
Mr Yousaf said: “This week we will discuss how we put together a viable public sector bid that could be ready for 2020.”
Labour and the three main rail unions have called instead for re-nationalisation of ScotRail - or direct state control by ministers.
The pressure on ScotRail and Mr Yousaf intensified on Thursday after a train broke down in Edinburgh, leading to the cancellation or part-cancellation of 225 others - some one in ten ScotRail services that day.
The minister said the only calls for his resignation had been from Labour and Labour-affiliated unions.
Scottish Labour transport spokesman Neil Bibby said Mr Yousaf would do everything possible to avoid taking responsibility for the "chaos" on Scotland's railways.
"Instead of questioning the motivations of trade unionists working on the railways, the minister should hold his hands up and accept responsibility for his failure to deliver a railway network that passengers in Scotland deserve.
"When the SNP awarded the contract to run the railways to Abellio, ministers said it was a 'world leading' deal, but now today Humza Yousaf has admitted the performance has been anything but.
"Overcrowded trains, delayed trains and cancelled trains: that’s the SNP’s record under Humza Yousaf.
"Passengers are fast losing confidence in him - the clock is ticking.
"Scottish Labour have said we need a People's ScotRail that puts passengers before profits, and are happy to help the transport minister understand the benefits of this.
"However, here and now Humza Yousaf needs to focus on making sure his improvement plan makes services better, not worse, and that he has a plan for an operator of last resort should the Abellio franchise fail."