In his first major policy proposal, Thompson challenged presidential rivalsand by criticizing "sanctuary cities" where city workers are barred from reporting suspected illegal immigrants who enroll their children in school or seek hospital treatment.
"Taxpayer money should not be provided to illegal immigrants," Thompson said at a round-table discussion that included Collier County, Fla., sheriff Don Hunter.
Thompson has argued his rivals are soft on illegal immigration because Giuliani, as New York mayor, sued the federal government to keep his city's sanctuary policy and because Romney tolerated sanctuary cities as Massachusetts' governor.
In turn, Giuliani's campaign accused Thompson of being weak on the issue. At a news conference Tuesday in Boston, Giuliani said: "I'm the one who can bring about immigration reform."
The immigration issue is important to many conservatives who influence Republican primaries. Some argue that illegal immigrants are straining schools and hospitals and taking jobs from U.S. citizens.
Thompson chose to announce his plan in Collier County, which has vast tomato farms that hire thousands of immigrants and last year was part of a two-county sweep that saw 163 illegal immigrants arrested in one weekend.
Thompson's campaign said 22 percent of the county's crime is committed by illegal immigrants.
To the sheriff, Thompson said: "You've clearly been swamped with a particular kind of problem because the federal government, in large part, has let you down and has not done their part."
"There's not a lot of new legislation that needs to be passed," the candidate said. "We need to enforce the laws that are on the books. There are laws against illegal immigration, there are laws to secure the border, there are laws against sanctuary cities, there are laws against publicly funding illegals, and that law is being disregarded."
Under Thompson's plan, sanctuary cities would lose discretionary federal grants, as would colleges and universities that allow illegal immigrants to pay in-state tuition.
The former Tennessee senator also called for stronger laws forcing employers to verify that workers aren't illegal immigrants, for a more rigorous system to track who is coming in and out of the country and for increased prosecution of "coyotes," smugglers who bring illegal immigrants across the Mexican border.
Calling for stronger border security, he said: "A small amount of nuclear material could do a lot of damage in the wrong hands. It makes you wonder why a terrorist would bother going through an airport or a port ... when we have an open border." Thompson's proposal calls for doubling the number of border patrol and customs agents.
"In 1996 we passed a bill, I was in the Senate, that outlawed sanctuary cities. Mayor Giuliani went to court to defeat that law," Thompson said.
Giuliani spokeswoman Katie Levinson said Thompson didn't try to fix the problem of illegal immigration when he was in the Senate.
"He was voting against $1 billion to combat illegal immigration at the borders, against stricter employment verification and for giving illegal immigrants more benefits than we give legal immigrants. That's not consistent or conservative," Levinson said.
Romney spokesman Kevin Madden called Thompson a latecomer to the issue of sanctuary cities. "Governor Romney has been the strongest candidate when it comes to demanding that our existing immigration laws are enforced," Madden said.
Romney has spent several weeks criticizing Giuliani for New York's sanctuary policy; Giuliani responds that he cracked down on all lawlessness and that Romney tolerated sanctuary cities in Massachusetts.
Romney says he tried to curtail the problem by deputizing state police to enforce federal immigration laws.
Romney and Giuliani both are calling for tougher border security and enforcement of immigration laws, although in the past they spoke favorably of measures, sponsored by Arizona Sen., another rival, that would provide a path to citizenship for the estimated 12 million immigrants here illegally.
The leader of a pro-immigrant advocacy group told CBS News' Nancy Cordes said Thompson's plan was not particularly original, but was not far off from what some strong opponents of illegal immigration, like presidential candidate and Rep., R-Colo., have proposed.
"Nothing new here that hasn't been proposed by others or by House Republicans," he said. "On a scale of 1 (Kennedy) to 10 (Tancredo) he is about an 8."
As Thompson traveled through Florida, his campaign suffered another staff departure and lost a key supporter in New Hampshire to rival John McCain.
Aides confirmed that Nelson Warfield, a political media strategist, has left the campaign while New Hampshire Republican Dan Hughes said he had joined McCain's campaign leadership team.
Hughes, a developer who helped Ronald Reagan's campaign in 1980 and later Reagan's White House transition team, publicly split from the Thompson campaign last week, citing Thompson's lack of interest in the early voting state.