There are hundreds of them - the ''criminal Marielitos'' they are called - and they have spent almost every day in prison since they arrived in the United States from Cuba three years ago.
Now Congress thinks it sees a way to send them home, back to Fidel Castro, back to the Cuban jails from which most of them came.
The White House is being urged to strike a bargain with Castro: If Cuba wants to get the men back who were captured during the fighting on Grenada last week, then Havana must also take back the criminal Marielitos now languishing in US prisons.
It's an idea that has drawn wide support on Capitol Hill. The Senate passed a resolution urging President Reagan to pursue the proposal. And in the House, 227 members have signed a letter backing the idea.
But a number of officials who have studied the problem of the criminal Marielitos for years say the idea would be difficult - and risky - to carry out.
The whole issue has gained urgency in recent weeks. An Atlanta federal judge, Marvin H. Shoob, has ruled that the federal government cannot keep the Marielitos in prison indefinitely - no matter how serious their crimes may have been back in Cuba.
His ruling is being appealed. If it stands, however, the result could be that a large number of Cuban prisoners convicted of very serious crimes could be quickly walking the streets in the United States.
The entire problem has its roots back in 1980. That year, 125,000 Cubans fled their island nation in sailboats, fishing boats, pleasure craft, and just about anything else that would float. They landed on Florida's shores to seek a new life.
The Castro government, which permitted the exodus, saw an opportunity when it realized that President Carter was going to allow an almost unlimited number of Cubans into the US. Castro took some of his country's most hardened criminals, people convicted there of murder, rape, and other heinous crimes, and mixed them in with the fleeing Cubans.
Many of these criminals slipped into US society undetected. Others, however, were screened out by US officials and jailed. They have been there ever since. Their numbers are estimated at about 550.
The number in prison has continued to rise, however. Since their arrival here , perhaps 2,000 Cuban boat people have been convicted of crimes committed in the United States. That is supposed to be grounds for deportation. But Castro will not take them back.
Altogether, including the criminals from Cuba and those convicted of crimes in the US, there are now 1,089 Marielitos at the federal prison in Atlanta and 26 at other federal prisons. Another 73 are in a mental hospital in Washington, D.C. And more than 1,000 are believed to be in state and local jails and prisons across the country.
The cost of locking up and treating these people is currently costing federal taxpayers more than $25 million a year. The total tab to Washington for all the Cuban boat people, including welfare and other costs of settling the 125,000, has now risen to more than $1 billion.
The Senate resolution, passed easily by voice vote, urges President Reagan to ''insist that as a condition for repatriation of Cuban nationals captured by US armed forces in Grenada the government of Cuba should agree to the return to Cuba of all Cuban nationals in the US who are found to be deportable under the immigration laws of the United States.''
Would the idea work? Rep. Bill McCollum (R) of Florida, who sponsored the House letter to the President, thinks it could. ''Castro is not going to endanger the lives of the 600 Cubans captured on Grenada,'' he says. He suggests there are number of ways of delivering the prisoners to prevent Castro from welshing on a deal.
The military prisoners, for instance, could be mixed with the criminal Marielitos and sent on a ship back to Cuba. Or they could all be sent together through the gate at the US naval base at Guantanamo, Cuba.
Critics of the idea note several potential problems: First, what if Castro turns the proposal down? Would the US then keep the 600 Grenada prisoners? Second, it might set a dangerous precedent if the US pushed the prisoners out the gate at Guantanamo. Would Cuba try to push people it didn't want right back in? Third, would this action upset other agreements with Cuba, such as the one on airplane hijackings?