President Bush has urged all sides to tone down their rhetoric
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The full US Senate is due to debate an immigration reform bill which is more lenient than tougher legislation passed by the House of Representatives.
The Senate Judiciary Committee voted on Monday on a bill that would legalise undocumented workers, backing White House plans for a guest-worker scheme.
The House of Representatives has backed a border security bill that sought to criminalise illegal immigrants.
Any legislation passed by the Senate will have to be reconciled with it.
The legislation voted by the House in December has been attacked by some Republican sectors, Democrats and pro-immigrant groups.
The bill includes the use of troops and police to halt migrants and tighter employment controls, as well as the creation of a 1,130-km (700-mile) fence along parts of the Mexican border.
There were more demonstrations on Tuesday over the House proposal to criminalise undocumented workers. Thousands of students in California and Texas walked out of classes for the second day in a row.
About half a million people rallied in Los Angeles, California, at the weekend.
Divisions
The Senate Judiciary Committee added two provisions to the immigration bill.
It voted down proposed criminal penalties on immigrants found to be in the country.
And, it also agreed to ultimately grant them citizenship - provided that they hold jobs, pass criminal checks, learn English and pay fines and back taxes. Senators said this did not amount to an amnesty.
The guest-worker scheme voted by the committee would allow foreign workers to stay for a set time in specific jobs.
The Republican Party has been divided over this particular issue.
President George W Bush has called for a "civil and dignified" debate. He said that securing borders was a top priority, but he also invoked the country's history as "a nation of immigrants" to argue for a balanced approach.
He argued that a guest-worker programme would provide a legal way of matching immigrant workers with employers, giving them jobs that Americans did not want.
It is estimated that 11.5 million people are living in the US illegally.
Many of them work in the agricultural sector and the construction and service industries.