Mr Wishart, the MP for Perth and North Perthshire and the SNP’s parliamentary business spokesman, posted on twitter on Tuesday that he wanted to be “the first post-war Speaker to emerge from beyond the two main parties”.
In a manifesto for the role, Mr Wishart said it was “time for a Speaker to come from beyond the big two parties to demonstrate that Parliament represents all shades of political opinion and all parts of the UK”.
Twitter user Caspar10666 replied: “You’re not settling in Pete, your supposed to be settling up. This will not go down well.”
SophiaPangloss posted: “I can only take this as a message Pete, either that you're not taking this seriously, or that you're telling us that the party is not being serious.”
And ChrisMcCusker67 responded: “I do not believe that any of our MP's should become part of the establishment whatsoever. All of you should come back home now, not become what I hate.”
The current Commons Speaker, John Bercow had promised to step down this year after a decade in the role, and reports earlier this year suggest he told friends he was ready to quit in the summer.
However, Mr Bercow is also understood to want to remain in the Speaker’s chair until Brexit has been resolved, with the UK’s exit date now delayed until October.
In his manifesto, the SNP MP promised to introduce electronic voting at Westminster to “end the massive waste of time in how we vote in the House”, and said he would end the practice of MPs being called to speak in debates based on seniority.
In a further break with centuries-old tradition, he also promised to stop calling MPs by the name of their constituency, and to abolish the dress code that requires men to wear jackets in the Common chamber. Mr Wishart said MPs would be allowed to clap in the chamber under his Speakership.