But she acknowledged the negotiations were going to be tough after a detailed account in the German press of their meeting suggested Mr Juncker left fearing the negotiations would end in failure.
In an interview with the BBC, Mrs May turned to the description of her coined by former cabinet colleague Ken Clarke.
She said: "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."
Mrs May added: "Well these are going to be tough negotiations as we as we go ahead.
"I'm asking the British people to give me a mandate to go into those negotiations.
"Every vote for me and my team is a vote to strengthen our hand in ensuring that we get the best possible deal for the United Kingdom."
Asked whether Mr Juncker had told her at the dinner "Brexit cannot be a success", Mrs May said: "I don't recall the account that has been given of the meeting that took place.
"I think a lot of this is Brussels gossip."
She said the UK needed the "right leadership" in the negotiations: "The 27 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."