The surprise move comes as sources had told CNN Corker was being "very seriously" vetted by the Trump campaign.
But Corker, who is chairman of the Senate Foreign Relations Committee, told CNN he is instead open to serving in Trump's cabinet, perhaps as secretary of state, and will speak at the Republican convention later this month.
"I feel like I'm better suited to other kinds of things and I think there are probably better suited people for this particular job for now," Corker told CNN on Capitol Hill.
Corker declined to say whether he and Trump had discussed particular positions in a Trump administration.
"I don't want to get into that so much," Corker said. "I think he knows who I am and I think he knows the kind of person who I am."
Corker also suggested a running mate for Trump: his daughter, Ivanka.
"His best running mate, by the way, would be Ivanka," Corker said. "I know that would not pass muster, but she's most impressive."
Two sources familiar with the vetting process told CNN earlier Wednesday that Corker was "very seriously" being vetted and had submitted a lot of information to Trump's campaign. He was on a very short list, the sources added.
But although he had received paperwork from the campaign, he had not turned it all over, a source familiar with the process told CNN. The campaign had yet to ask him for all of the paperwork, and Corker had not had his final interview with Trump regarding the position, the source added.
Corker campaigned with Trump in Raleigh, North Carolina, on Tuesday. And he told CNN's John Berman on Wednesday evening that he felt the Trump campaign was turning a corner in terms of its professionalism.
"You're going to see a lot of differences moving ahead," Corker said, voicing his displeasure with the "caricature" that he said had been sketched of Trump. "They're making the adjustments that they need to make to rise to that occasion."
Also on Wednesday, Iowa Sen. Joni Ernst, another potential Trump running mate, downplayed her prospects, telling CNN she has not received documents from the Trump campaign for potential vetting. But she said she was open to speaking at the convention.
"Well yes, I would speak," Ernst told CNN outside her office on Capitol Hill. "I have offered my services and I think it would be a nice opportunity."
Later Wednesday, a Republican adviser confirmed to CNN that Ernst will be speaking at the convention on the topic of national security.
Pressed on whether she has discussed being Trump's running mate with the candidate himself, Ernst deflected, saying they discussed Iowa's importance in the upcoming election.
Just two weeks from the Republican National Convention, Trump's vice presidential search -- and speculation about who he'll pick -- is at a fever pitch.
Speaking on Fox News Wednesday afternoon, Trump called Corker a "terrific guy" and said he was looking at 10 candidates, including some military generals.
"I like the concept of the generals," Trump said. "There are two of them under consideration."
Previously, Trump has said he wants someone with political experience to help him pass his legislative agenda through Congress.
Trump met in person over the holiday weekend with Ernst and Indiana Gov. Mike Pence. And Trump's recent praise of Arkansas Sen. Tom Cotton has recently attracted interest.
Former House Speaker Newt Gingrich, who has discussed
what an ideal vice presidency would look like to him, will campaign with Trump Wednesday in Ohio. And New Jersey Gov. Chris Christie is believed one of the top names on the short list for the number two spot.
Christie will travel with Trump to Miami on Friday, Trump spokeswoman Hope Hicks told CNN.
But Trump's campaign has been quiet about who he'll pick -- and when he'll make an announcement.
"It's really going to come down to the personal relationship that he and Mr. Trump could have to find out if they could work well together," Trump's former campaign manager Corey Lewandowski said Sunday on CNN's "State of the Union."
, now a CNN political commentator, emphasized the importance of Trump tapping a politician with ties to Washington, who could help advance his legislative agenda in Congress.
"Having someone who has those relationships in Washington is going to be a critical component of his presidency," Lewandowski said.