Instead, Wolf has held back.
His decision - which faces a deadline this week - could prove a make-or-break moment for his administration, not just for the year but for the rest of his term.
"It's an unenviable position to be in right now," said Chris Borick, political science professor at Muhlenberg College in Allentown. "All of his options have really unappealing sides to them."
Last week, the governor was clear that he intended to use his veto power to strike $6.6 billion out of the $30 billion spending plan sent to him by the legislature, much like he did twice last year, extending the historic impasse.
The latest budget contains $200 million more for public education - a priority for Wolf - but would not hike the state sales or income tax as he has proposed.
Wolf wants the tax revenue to help close a gaping deficit and end the annual wrangling with the legislature over how to keep funds flowing for critical state programs. Republicans have staunchly resisted any tax hikes.
"Republican leaders are once again insistent on passing another irresponsible and unbalanced budget that does not fund our schools or fix the deficit," the governor said last week.
This week, he faced a sobering reality: GOP legislators might be able to cobble together enough votes to override a veto - a two-thirds majority in both chambers is needed - with members of Wolf's own party tipping the scales against him.
Publicly, at least, Democratic legislative leaders say they are united. Privately, they have met with the governor several times this week, urging him to back off his threat of a full veto and consider at least releasing some money to public schools, as he did last December to ease the budget crunch.
The state has been operating without a complete budget since July of last year.
As they did last fall, many school districts have taken out loans to continue operating - but that financial lifeline, too, may soon wither.
"There are some school districts that are in some very desperate straits," Jim Buckheit, executive director of the Pennsylvania Association of School Administrators, said Tuesday.
Because of the impasse, the credit rating for the state's intercept service program, which guarantees school debt, has been downgraded, making it harder for districts to borrow money.
School districts that have taken loans also worry that if the budget is not resolved, they will not be able to repay loans by the end of the school year on June 30. "Then they would basically be in default," Buckheit said.
Democrats in the legislature have heard those calls - and have pressured Wolf to back off his promise of a full veto. Lurking in the background is the possibility that some could side with Republicans to override a veto - a potentially fatal political blow to the governor's agenda.
"You can't lose your own party, even if they are in the minority," said Philadelphia Democratic consultant Larry Ceisler, noting that such a predicament helped shatter Republican Gov. Tom Corbett's quest for a second term. "Tom Corbett lost his own party and look what happened to him. It is a marriage, for better or for worse."
In an interview Tuesday, Senate Minority Leader Jay Costa (D., Allegheny) acknowledged it "would be tough to keep our members together."
But, he added: "I'm confident, with work, that we will be able to keep our members in line."
Costa said it is more important to fight for Democratic platforms in the budget Wolf has proposed for the next fiscal year - the one that begins in July. In that proposal, the governor again asked for a tax increase to close a $1.8 billion deficit and support a major boost for public schools.
"If we don't do this in the right way," said Sen. Vincent Hughes of Philadelphia, the ranking Democrat on the Senate Appropriations Committee, "the same folks who strung out the 2015–16 [budget] could possibly string out 2016–17."
One fear is that another veto by Wolf will further alienate Republicans and strain relations, Hughes said. That could leave Pennsylvania without a budget not just for this fiscal year, but for the next.
"We're all looking pretty bad right now," he said.