The direct and unusual veto threat from Herbert comes as budget talks between the House and the Senate appear to have blown up over funding for Lockhart's legacy project.
Lockhart, is seeking $200 million to $300 million over the next few years to fulfill her vision to provide a digital-learning device to every one of Utah's more than 600,000 students.
Senate Republican leaders have said they could provide $26 million for the program.
Lockhart said lawmakers are still in the mid-point of budget negotiations and said "we can do a lot with more than $30 million."
"[The initiative] is a vision of moving us out of the 19th Century in education and transforming our system to better prepare our children for now and the future," Lockhart said. "I hope [the governor] will catch that vision and be a part of it."
House Majority Leader Brad Dee, R-Ogden, said constitutionally, it's not the governor's place to decide how much funding goes to programs.
"We budget. That's what we do, and we approve the amount of money that goes to specific programs throughout the budget. That is our purview, our constitutional responsibility," Dee said. "You can veto a bill on principle, but to veto a bill if it's over a certain amount, that's just saying I'm going to veto a bill if it's not my idea."
Showdown • The stand-off and Herbert's threat are the latest flare-up this session between Lockhart and the governor. Lockhart, who is not seeking re-election in November but hasn't ruled out running against Herbert in 2016, criticized the governor as an "inaction figure" on the opening day of the session and the two have traded barbs since.
Under Utah law, the governor has line-item veto authority, which allows him to wipe out specific appropriations that he opposes. That means Herbert could wipe out the speaker's pet project with a stroke of his pen if he deemed it was funded at a higher amount than it should be.
On Thursday, House leaders did not show up to a scheduled budget negotiation with Herbert and Senate leaders and Senate President Wayne Niederhauser said the House is not talking to the other parties.
Lockhart said the House notified the governor and senators they were not ready to discuss details of the budget plans.
Senate budget chairman Lyle Hillyard, R-Logan, said it is possible that the Legislature could adjourn next week without passing a budget something Hillyard said has not happened in his nearly three decades in the body.
"We've set forth our proposal and they theirs and we're quite a ways from agreement," Hillyard said. "If we don't come together, then I think what will happen is the governor will probably have to call us into special session."
The Legislature has already passed and Herbert has signed base budget bills, meaning the state government will continue to function at existing levels. Herbert would then have to call a special session at a later date to pass a complete budget for the next fiscal year.
Herbert said that would be an unfortunate outcome.
"We've got to get the budget [done]," the governor said. "We're starting to look a little bit like Washington, D.C., that can't even come together and even put together a budget. I'm optimistic and hopeful the Senate and House can come together and have the dialogue and discussions and put together a rational and reasonable and responsible budget."
The art of the deal • Lockhart said she has been in House leadership for 12 years and in many budget negotiations, and chances are "extremely slim" they won't strike a deal.
"I've been in leadership meetings where we have discussed the budget where one house would get up and walk out. That hasn't happened yet," she said.
Senate Republican leaders drew a sort of line in the sand Wednesday night, saying they would be willing to put $26 million into the speaker's education technology initiative. If Lockhart wants more, they said, it would require a tax increase.
There are two tax increases on the table one that would raise property taxes and boost the ongoing funding to Lockhart's plan to $50 million. The other would increase gasoline-tax revenue over a period of a few years.
House Republicans, who have staked out a position opposed to tax hikes, discussed the gas-tax proposal in a closed caucus Thursday afternoon.
"I find it interesting that the Senate would support a property tax increase when there are other places in the budget to find money to fund education," Lockhart said earlier this week. "If public education is our highest priority, then we will be able to find the money in other places."
Lockhart has said money set aside for transportation projects is a logical place to look for more funding for her technology program.
"We've been very good to transportation through the years," she said Tuesday. "We believe that there is revenue in transportation that we can redirect into public education that will not materially affect the transportation program in any kind of significant way that would cause problems."
The governor said his budget priorities remain the same as they have been since the start of the session providing $66 million to pay for more than 10,000 new students entering Utah schools next year; $62 million to increase Utah's per-pupil spending by 2.5 percent; and $20 million to balance state funding to the colleges and universities.
"So I can't see a way to get [the speaker's technology initiative] past 25 to 30 million dollars and frankly I'd have to consider a veto because it just won't work," Herbert said. "You're going to have to take away from so many other places you don't have the appropriate balance in the budget. It would not be responsible. It would not be reasonable."