Boris Johnson has urged European Commission president Jean-Claude Juncker to visit Britain to see how it has been "damaged" by the EU.
Mr Juncker invited the Tory Brexit campaigner to come to Brussels after claiming that the attacks on the European Union made by Mr Johnson during the Brexit referendum were not "in line with reality".
Mr Johnson said he would take up the offer but added it would be "wonderful" if Mr Juncker came to meet families and businesses in the UK that have been hurt by EU membership to give him a "better understanding" of the impact of EU membership.
Mr Juncker was questioned at the G7 summit in Japan about the former London mayor's comparison of the EU's efforts to unify Europe to earlier attempts by Napoleon and Hitler.
He said: "I'm reading in (the) papers that Boris Johnson spent part of his life in Brussels. It's time for him to come back to Brussels, in order to check in Brussels if everything he's telling British people is in line with reality.
"I don't think so, so he would be welcome in Brussels at any time.''
In response, Mr Johnson wrote to the EU's senior official saying: "You very imaginatively suggested today that I might like to visit Brussels, so that I could better inform myself of your work and your plans for the future development of the EU. I would like to take you up on your suggestion.
"As you may know, I believe that the time has come for a real and thoroughgoing reform of Britain's relations with the European Union and that the only way we in this country can take back control of our democracy is to vote leave on June 23.
"I believe that we should develop a more harmonious and practical relationship between Britain and other European countries, and that this should be done on the basis of free trade and intensive intergovernmental cooperation. I would much welcome the chance to explain how this would be of benefit both to this country and the rest of the EU.
"I would also like to extend an invitation to you. Many parts of Britain - many families and small businesses - have been damaged by our EU membership. It would be wonderful if you could visit some of these places and meet some of those people with me. I have no doubt it will help inform the debate and give you a better understanding as you attempt to reform the machinery of the EU.
"As you mentioned, I have happy memories of the beautiful city of Brussels and I look forward to seeing you in the near future."
Before his political career, Mr Johnson spent several years in Brussels as EU correspondent for the Daily Telegraph, where his articles were noted for their Euroscepticism.
The level of irritation at the Commission over the former London mayor's trenchant criticism was revealed in a tweet by a senior aide to Mr Juncker, who classed Mr Johnson alongside Donald Trump and French National Front president Marine Le Pen in a "horror scenario" line-up of potential world leaders.
Martin Selmayr, took to Twitter to argue that the prospect of Mr Johnson in Downing Street made it "worth fighting populism".
Envisaging a future G7 summit attended by Mr Trump as US president, Mr Johnson as British premier, Ms Le Pen as French president and maverick Five Star Movement leader Beppe Grillo as Italian PM, Mr Selmayr wrote: "#G7 2017 with Trump, Le Pen, Boris Johnson, Beppe Grillo? A horror scenario that shows well why it is worth fighting populism. #withJuncker."
Mr Selmayr is Mr Juncker's head of cabinet and his tweets go out under the banner "@EU_Commission #TeamJunckerEU", though he stresses that they are written "in a personal capacity".