The Prime Minister made the threat in an interview amid the on-going row, sparked by leaks from a private meeting between the Prime Minister and the EU Commission President.
Ms May also used the interview with the BBC to confirm that she intends to serve a full term to 2022 if she wins the election on June 8.
It comes after a German newspaper reported that in the wake of a meeting with the Prime Minister, Mr Juncker phoned Angela Merkel and said Ms May lives “in another galaxy” and is “deluding herself” over Brexit.
Responding to questions over the reports, she said: “I think what we've seen recently is that at times these negotiations are going to be tough.
“Now during the Conservative Party leadership campaign I was described by one of my colleagues as a bloody difficult woman.
“And I said at the time the next person to find that out will be Jean-Claude Juncker.”
The description of Ms May was first made by ex-chancellor Ken Clarke during last year’s Tory leadership contest, but the soon-to-be successful candidate later took ownership of it.
Leaks to the Frankfurter Allgemeine newspaper suggested EU Commission President Mr Juncker knocked back Ms May’s plan to reach a deal guaranteeing citizens’ rights as early as June, because he believed the issue too complex to agree so soon.
Downing Street said it did not recognise the version of the meeting reported by the publication.
While Ms May’s officials described the face-to-face as “constructive” just after it ended, Mr Juncker is reported to have said: “I leave Downing Street 10 times more sceptical than I was before.”
Speaking about the issue to the BBC, Ms May said: “What is I think possible and what we want to do is to be able to give reassurance to EU citizens living here and to UK citizens living in Europe.
“I've always said that there are complexities to this issue and lots of details that will need to be agreed. What people want to know is to have some reassurance about their future.
“I believe we can give that at an early stage. I've got the will to do this.”
Directly addressing the stories about the meeting she repeated her dismissal of them as “Brussels gossip”. She went on: “But what is important is that there is a key question for people when they come to this election.
“We've seen from all of this that these negotiations at times will be tough. Getting the right deal requires the right leadership. And there's only going to be one of two people sitting around that table. The twenty seven other EU countries on one side of the table and who is going to be there standing up for the UK? It's either going to be me or Jeremy Corbyn.”
Confirming her intention to serve a full term, she said: “I have no intention of doing anything other than serving the full term until 2022 because this is, as I say, an important time for our country and what we do over the next five years could change our country for the better for the future and truly make it a country that works for everyone, not just the privileged few. That's what I'm in it for.”