An Early Halloween Fright for Wall St.: Elizabeth Warren for Treasury Secretary?

By Sandra Sobieraj Westfall

10/22/2014 AT 11:50 AM EDT

Millions have viewed her now-famous C-SPAN snippets on YouTube with tantalizing – for Senate hearings – titles like "Elizabeth Warren EMBARRASSES Bank Regulators at First Hearing," or "Elizabeth Warren: Classic Takedown…" and supporters are already lining up to back an "Elizabeth Warren for President" campaign in 2016. But is the freshman senator from Massachusetts herself on board with a run for the White House? Warren wrinkles her nose.

"I don't think so," she tells PEOPLE in an interview conducted at Warren's Cambridge, Massachusetts, home for this week's issue. "If there's any lesson I've learned in the last five years, it's don't be so sure about what lies ahead. There are amazing doors that could open."

She just doesn't see the door of 1600 Pennsylvania Avenue being one of them. Not yet, anyway. "Right now," Warren says, "I'm focused on figuring out what else I can do from this spot" in the U.S. Senate.

The former Harvard Law School professor whose aggressive oversight of the financial industry after the 2008 economic meltdown not only won her a national reputation as watchdog for average Americans and swept her into national office in 2012, but turned her into an Internet sensation.

But when she reviews the clips herself, Warren second-guesses whether she's being tough enough. "I watch and think, 'Damn! I should have gotten that question a little sharper,' " she admits with a laugh.

But all that makes for click-bait on YouTube also makes her a lightning rod on Wall Street.

"They're scared to death of Elizabeth Warren because she knows not only how banks operate, but she knows all about anti-trust law," says Tufts University political science professor Jeffrey Berry.

So, if not chief executive, how would Warren feel about being Treasury Secretary under a President Hillary Clinton? Warren belts out a big, Clinton-caliber belly laugh. "Well, THAT'S a fun thought!"

As for Clinton, Warren says of the two women's relationship: "We have talked. It's not much more than that. Not much more."

For more of PEOPLE's at-home interview with Warren, pick up a copy of the new issue of PEOPLE on newsstands Friday