The bill would not touch the structure of the US visa system or reform any of the legal immigration laws in the US, but the GOP senators backing it say border security must come first before a broader immigration measure goes forward.
But Cornyn emphasized that he viewed the bill as lawmakers leading on border security, not the way the White House has approached the issue.
He said the bill would be the opposite of a "one-size-fits-all solution" and emphasized that he favored it over an "ad hoc" approach.
"It represents a plan that now Congress can then implement through the appropriations process rather than appropriating money without a plan in place on a piecemeal or ad hoc basis," Cornyn said, taking a shot at the individual requests the administration has made to fund particular stretches of border barrier.
Cornyn wouldn't rule out supporting a $1.6 billion request for 74 miles of wall that the House approved in his chamber, but said it wasn't his preference.
"We don't know how that $1.6 billion fits into that plan for the entire southern border. Doing this on a piecemeal basis, I think, is not the most efficient and most effective way to do it," he said. "I would prefer to have a plan in place, but absent that, we need to do what we have to do to secure the border."
And he also took steps to signal that Trump's oft-promised wall wasn't a "new" idea or the impetus for his interst.
"The idea of physical infrastructure along the border is not a new concept," Cornyn said, standing next to photographs of existing border infrastructure. "I like this particular chart ... it demonstrates that the idea of physical infrastructure or wall is not novel. President Trump didn't dream that up, it's something that exists."
All the lawmakers emphasized the bill should come before any other immigration reform efforts.
"Until our borders are fully secure, the current system will continue to reward people who enter our country illegally over those who follow the law," said Wyoming Sen. John Barrasso. "This bill will make sure our immigration officers have the tools and resources they need to enforce our laws and secure our borders."
The measure includes Kate's Law, which creates increased penalties for immigrants that repeatedly come to the US illegally after being deported. It also would punish "sanctuary cities" -- a catch-all term that describes jurisdictions who in some way don't fully engage in federal immigration enforcement -- by taking away federal funds.
The bill would also create new criminal penalties for transnational organized crime and would increase vetting measures for people seeking visas to the US.
In another potentially controversial measure, the bill would require that unaccompanied minors who cross into the US illegally are screened quickly to potentially allow them to be sent immediately back to their home country.
Unaccompanied minors are a focus of lawmakers who claim that the system of settling them with family and guardians in the US allows for the spread of gang violence -- though there is no statistical evidence to back up the concern. Currently, unaccompanied minors are turned over to the Department of Health and Human Services for placement with a guardian, in part due to court rulings limiting how they can be treated in federal custody.
The Senate requires 60 votes to move legislation forward, meaning at least eight Democrats would be needed on the bill.
Senate Majority Whip John Cornyn unveiled the bill Thursday with Barrasso, Wisconsin Sen. Ron Johnson and North Carolina Sen. Thom Tillis.
The legislation is similar to a border security bill
introduced in the House by House Homeland Security Committee Chairman Mike McCaul.
Cornyn did not say whether the White House was supporting the measure, but said his lawmakers worked with the Department of Homeland Security to get input on the development of the bill.
CORRECTION: This story has been updated to correct the cost of the bill.