Michel Barnier has insisted Brexit talks will start hours after the General Election, adding he is not aiming for "no deal or a bad deal".
The EU's chief Brexit negotiator also dismissed suggestions that the UK will be required to pay up to 100 million euro (£85 billion) as an "exit bill".
He said the UK's financial liabilities will be dependent on the date of Brexit and the methodology adopted, adding the European Council will "work with the Brits, very calmly" to reach an agreement.
Mr Barnier's remarks came after former Ukip leader Nigel Farage warned Britain could walk away from talks by the end of the year as a result of Jean-Claude Juncker's "bloody rude" behaviour and attempt to "bully the Brits".
The MEP said the EU needs to make "grown-up, reasonable" demands.
Mr Farage's attack came after European Commission president Mr Juncker said the Brexit process shows how "united and prepared" the EU will be.
European Council president Donald Tusk also insisted the conduct of the talks must "show the European Union at its best", adding this involves "unity, political solidarity and fairness towards the United Kingdom".
Mr Barnier, speaking as the European Parliament discussed the EU's negotiating strategy, said: "Our aim is to achieve an agreement.
"I certainly don't intend to have no agreement, no deal or a bad deal. We want to conclude a deal with the UK, not against the UK.
"In fact, I very much appreciate on the UK side you could find the same spirit to reach a deal with the EU, not against the EU."
Mr Barnier said he hopes to start the talks as soon as possible, adding: "The day after the elections that (Prime Minister Theresa May) has called for on June 8.
"Clearly, the fact that elections take place just before the start of the negotiations will give a clear stability and visibility to our UK partners.
"That's what we need as well and I think that's one of the key elements for these negotiations to be a success."
Mrs May and her ministerial colleagues have repeatedly claimed no deal is better than a bad deal from the UK's point of view.
Mr Barnier, on the financial settlement expected with the UK, said: "I'm pleased Mr Farage is here because he quoted me.
"Now, I've never quoted these figures. The figures will depend on the methodology that we have to agree on. It'll also depend on the actual date of the UK's exit.
"It's not (me) who is going to set a figure. The European Council has said we're going to work together with the Brits, very calmly, to find an agreement on commitments undertaken.
"I'm going to do it perfectly calmly. It's not about revenge, it's not about punishment, I'm simply going to look at the figures and facts and the commitments that have been undertaken - no more, no less."
Relations between Brussels and London have been strained further by leaks about a difficult dinner in Downing Street attended by Mr Juncker, which reportedly ended with him saying he was "10 times more sceptical" of the likelihood of a successful Brexit.
Mr Farage had earlier told fellow MEPs to imagine they were hosting a dinner party in which a "very important guest" had told others that you were "deluded" and "living in a different galaxy" within hours of it ending.
He described a situation in which the details of confidential conversations were "blabbed to an opposition newspaper", adding: "I don't know, Mr Juncker, whether this is how you carry on in Luxembourg.
"I doubt it, because in any part of the civilised world, frankly, that behaviour would be considered to be bloody rude and the act of a bully.
"Well, I'll tell you something, your attempt to bully the Brits through this negotiation is not working - 68% now of the British people want Brexit to happen.
"All of that on top of a ludicrous ransom that Monsieur Barnier wants, that I'm told has now doubled to 100 billion euro.
"Either we get some grown-up, reasonable demands from the European Union or the United Kingdom will be forced to walk away before the end of this year.
"We can't spend two years with this farce."