(CNSNews.com) - Texas Republican Governor George W. Bush has said all he intends to say about rumors that he used illegal drugs and from this point on will consider the question answered, according to his campaign.

Asked what Bush's response will be in the future to the questions surrounding rumors that he has used cocaine in the past, campaign spokeswoman Mindy Tucker told CNSNews.com his response will be, "I've addressed that."

In a statement released Thursday Bush said he will not play what he called the "game of responding to rumor and innuendo," which he said prevents quality people from running for public office.

"This game drives good people away from the political process and I will not be part of it. The American people are sick and tired of the game of trying to destroy someone's good reputation based on ridiculous gossip," Bush said in his statement, adding that rumors building in the past few weeks "have been greatly exaggerated."

Reporters continue to question the candidate publicly about unsubstantiated rumors of drug use. But one observer thinks he may be able to shed some light on what has powered the rumor mill.

University of Virginia political scientist Larry Sabato, appearing Wednesday on CNN's Inside Politics, said he has been approached by two "national reporters" who indicated the rumors were coming primarily from two rival Republican presidential campaigns.

Sabato told CNSNews.com that the reporters had been encouraged by rival campaign staffers to press Bush on the issue.

Bush criticized reporters Wednesday during a press conference in Austin when they continued to hammer him about unsubstantiated rumors that he used cocaine years ago.

"You know what happens? Somebody floats a rumor and then it causes you to ask a question. And that's the game in American politics, and I refuse to play it. That is a game, and you just fell for the trap. And I refuse to play," Bush responded.

Later, the GOP presidential contender responded to a question on whether he would enforce background checks for White House staff - and whether he would submit himself to the same standard.

"My answer is absolutely," Bush said. "I have told the people of this country that over two decades ago I made some mistakes, when I was younger. I have learned from those mistakes.

Bush continued, "Should I become the president, my pledge to the American people is that I will uphold the honor and dignity of the office to which I have been elected," an allusion to the series of scandals which have plagued the current administration.

News reports and talking-head shows continued discussion on the issue throughout Thursday while Bush conducted a photo-op in Roanoke, Va.

As of Thursday, no one has formally accused Bush of having used drugs of any kind.