Owner Jerry Reinsdorf has been focused on an attempt to move the team to the South Loop with the help of funding from the state. Pritzker appeared to balk at that idea.
"A billion dollar investment in a private business- for a private business, rather, that's owned by wealthy people and is a highly-valued enterprise, it seems like the taxpayers, rightly, should say, 'Don't we have other uses for that capital?'" the governor said.
The Sox are also once again the subject of rumors involving the possibility of selling or relocating the baseball team.
Fans and the owners of Grandstand on the South Side said on Friday that the new stadium talks only make them more frustrated.
"Build the team! We don't need a new stadium. We don't need more money draining out of the city. We need the team," said Grandstand owner Stephanie Ganal.
The White Sox refused to comment this week on the possibility of selling or relocating, but there are interested ownership groups who would like to buy the team and possibly move it to Nashville.