Cornyn made his views known yesterday on Capitol Hill during a meeting of the Senate Judiciary Committee.
He cited Holder's failure to cooperate with a subpoena about the justice department's gun-smuggling project, Operation Fast and Furious.
Thousands of weapons fell into the hands of Mexican drug cartel members because of the operation.
Plus lawmakers say Holder did not comply with a subpoena stating Holder had to appoint a special independent prosecutor to look into recent national security leaks.
Congressman Thornberry told us he disagrees with a lot of the actions Holder has taken, but says he recommends a different solution than Cornyn's.
Senator John Cornyn said, "So, Mr. Attorney General, it is with sorry and regret more than anger that leads me to say that you leave me no alternative but to join those who call upon you to resign your office."
Congressman Thornberry says, "From the beginning of his tenure, there have been a number of bad judgments that have been made. I think, though, the way to solve this is with the ballot box in November and electing a new administration."
Holder risks being found in contempt of Congress and could face severe penalties including jail time.
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