Republican presidential hopeful Rick Santorum sought to remind voters Thursday that he has been calling for stricter immigration laws long before any of his rivals in the nomination race, as he rolled out a proposal that calls for scrapping birthright citizenship and curbing legal immigration.

Mr. Santorum, who represented Pennsylvania for two terms in the U.S. Senate and ran for president in 2012, said he welcomes the gravitation of Donald Trump and Wisconsin Gov. Scott Walker toward a message he said he has been delivering for years about enforcing laws on the books and putting American workers first in immigration decisions.

“Until this summer, I was the only candidate who had a message focused on helping American workers by putting common-sense limits on this surge of immigrants,” Mr. Santorum said at the National Press Club in Washington. “Gov. Walker was first to change his position and with few specifics calls for limits on immigration.”

“Now Donald Trump has joined a majority of Americans and me with some ideas on helping to put American workers first,” he said.

Immigration has divided the Republican field, with Mr. Santorum, Mr. Trump and, to a lesser extent, Mr. Walker insisting that the party move beyond a “secure the border” stance and focus on American wages.

Mr. Santorum, the runner-up for the Republican nomination in 2012, is far behind his rivals this time but says voters should give him credit for having been on top of the immigration issue early.

Mr. Santorum said he would end the Obama administration’s “catch and release” and President Obama’s executive amnesties that have spared most illegal immigrants any danger of deportation. He said he would pull federal funding for “sanctuary cities” and deny visas to countries that refuse to take back their citizens.

The former senator called for a crackdown on immigrants who overstay their visas by using biometric identification for each immigrant entering and leaving the country.

In addition, Mr. Santorum vowed to secure the border through a combination of deploying more manpower, using the latest technology and building hundreds of miles of walls — though he tried to distinguish himself from Mr. Trump, who has said he would try to make Mexico pay for the wall.

“While I won’t demand the government of Mexico to build the wall — I want U.S. workers to do that — I will make it clear to the Mexican government that they must stop facilitating this lawlessness on the border and cooperate with our efforts,” he said. “However, if they fail to cooperate, I am prepared to stop authorizing border crossing cards as a first step in getting their cooperation.”

Mr. Trump last week offered a six-page plan that dovetails with much of the proposal Mr. Santorum called for Thursday, including a shared desire to end birthright citizenship for the children of illegal immigrants and reduce the number of legal immigrants coming into the country each year.

Mr. Santorum wants to reduce legal immigration by 25 percent.

“I believe immigration can be a very good thing, but as with anything, there can also be too much of a good thing,” Mr. Santorum said. “When our labor markets cannot manage the influx we are receiving, then it is time to recalibrate.”

He also took aim at Sen. Ted Cruz of Texas, another Republican candidate who has voted against legalization proposals in Congress, labeling them amnesty, but who has called for an increase in guest-worker programs and legal immigration.

“Some, like Ted Cruz, [have] actually proposed doubling legal immigration,” Mr. Santorum said.

During the 2013 immigration fight on Capitol Hill, Mr. Cruz opposed efforts granting illegal immigrants a path to citizenship and pushed an amendment to the “Gang of Eight” bill that would have increased the H1-B visas by 500 percent, according to his Senate website.

“Legal immigration is a fundamental pillar of our nation’s heritage, and I was pleased today to offer legislation that would have improved and expanded legal immigration by dramatically increasing the cap for high-tech temporary worker visas,” he said at the time. “There is currently a serious shortage of workers in the fields of science, technology, engineering, and math, yet every year we send thousands of high-tech graduate students back to their home countries to start businesses and create jobs. This makes no sense.”

The Cruz camp declined to respond directly to Mr. Santorum’s comments but did note that Mr. Cruz never proposed, as Mr. Santorum said Thursday, a “permanent work permit” for immigrants.