WASHINGTON (CNN) -- The U.S. Senate will take part in a vote of no-confidence on embattled Attorney General Alberto Gonzales after debate on the immigration bill is completed, Democratic Sen. Charles Schumer said Thursday.

The non-binding measure likely will be considered in mid-June, unless the immigration legislation runs into problems. The House plans to vote on a similar measure next month, after a congressional recess.

Senate Majority Leader Harry Reid authorized the introduction of a resolution requesting the non-binding vote, according to Schumer.

President Bush has repeatedly supported Gonzales -- the president spoke again Thursday of his confidence in the justice official. But Schumer said many people don't agree with him.

"I think we hoped that it would never come to this," the New York lawmaker said at a news conference. "We would have liked the attorney general to have stepped down on his own, but the rule of law has been trampled, confidence in the Department of Justice has been shattered and and leadership is virtually non-existent."

Most recently, the department Gonzales heads has come under scrutiny for the firing last year of eight U.S. attorneys, and is accused of making those decisions for partisan, not performance reasons.

"I think all of us have reached the conclusion that these firings were improper," said Sen. Dianne Feinstein of California.