"I was speculated about eight years ago, and it's nice to be speculated about," he said on MSNBC's "MTP Daily."

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"In my gut, eight years ago, I didn't think it was going to go that direction, and I got to say, in my gut right now, I kind of feel like I'm going to stay in the Senate."

Kaine said he expects to remain in the Senate and continue his work on armed services, foreign relations and budget issues that "make me a happy senator every day."

"Speculation is fine, but I got a job to do," he said.

"And for the Hillary campaign, the best thing I could do is hopefully help her win in Virginia. Virginia is a key state and I'm looking forward to campaigning for her to make sure she wins Virginia."

Kaine is under consideration to become Clinton's vice presidential pick and is viewed as one of the top contenders. But some have questioned whether the senator might be too centrist of a pick after a primary season where Clinton faced deep skepticism from the left.