Asked whether Mr Rumsfeld would stay at his post until he leaves office in early 2009, Bush replied: "Well, the end of my term is a long time, but I tell you, he's done a heck of a job, and I have no intention of changing him."
The president, whose comments came during an exclusive interview with Fox News Channel, also said he and Rove, who may face legal jeopardy in a CIA leak investigation, "are still as close as we've ever been."
"We've been through a lot. You know, when we look back at the presidency and my time in politics, no question that Karl had a lot to do with me getting here, and I value his friendship. We're very close," said Mr Bush.
The president was responding to a number of media reports saying that, in the face of slumping public opinion polls and rising discontent with the war in Iraq, his relationship to Mr Rove and Mr Cheney had chilled.
"The vice president goes through, I guess, what all people in Washington go through at some time or another, and that’s kind of a massive speculation about whether he’s running the government or not running the government, whether I like him or don’t like him," said Mr Bush.
"The truth of the matter is that our relationship hasn’t changed hardly at all. He’s a very close advisor. I view him as a good friend," said the president.
"I’d say the relationship, it’s only gotten better. We didn’t know each other that well when we first came to Washington DC, and my respect for him has grown immensely," said Mr Bush.