The Italian prime minister has branded the Brexit vote a "bad decision" and blamed it on David Cameron.
Matteo Renzi also suggested the UK will not be able to have full access to the single market while curbing immigration, saying it will be "impossible" for British people to have more rights than others outside the EU.
His comments come as former education secretary and Remain campaigner Nicky Morgan criticised the Government's "lack of a plan" and urged Theresa May to come up with a clear strategy in the next couple of months.
Speaking to the BBC, Mr Renzi said he was "shocked" when Britain voted to leave the EU but agreed that "Brexit is Brexit - the people of the UK decide the way for the future".
He said he "tried" to keep the UK in the EU, but blamed Brexit on Mr Cameron's decision to hold a referendum to solve his own internal party problems.
He said: "The problem was one problem. When David Cameron decided to use the referendum to solve the internal problems of the Conservative Party, this was the problem. We cannot use foreign affairs to solve internal problems.
"The decision of (the) British people is a bad decision in my mind and I'm sad for that, but if we don't accept the result of the referendum the risk is we give the message that a vote is not a good thing, democracy is not an asset for this continent."
He said he is ready to negotiate with Mrs May once she triggers Article 50 and starts formal talks on Britain's departure from the EU.
He added: "But it will be impossible to give British people more rights than the other people outside the EU."
Asked if there could be some "flexibility" over the EU's rules over freedom of movement and access to the the single market, he said: "This is a very interesting debate because this debate will be a debate about the concept of rules in the EU. But when the UK decides the opening of Article 50 we will discuss about this."
He urged Mrs May to trigger Article 50 "as soon as possible". Mr Renzi also stressed that he wants Britain and the EU to have a strong relationship following Brexit. He said: "Now the situation is that we can and we have to build the best alliance between the UK and the EU for the future because we will be best friends for the next years."
Mrs Morgan told the BBC Radio 4 Today programme that the balance between access to the single market and freedom of movement is "at the heart" of the Brexit issue, and urged Mrs May to hurry up and make her position clear.
She cast doubt over statements coming out of Europe suggesting there will be no budge on the issue, saying "nobody starts the beginning of the negotiating process where they end up".
But she urged the Prime Minister to make public her strategy for Brexit in the next couple of months, warning that if she does not then others will fill the void.
Mrs Morgan said: "At the moment there has been a lack of a plan from our Government about the negotiating the process, what's going to happen, what we are going to ask for.
"The longer that gap is left, the more likely it is that, as we are beginning to see, people are taking up positions whether it's hard Brexit or soft Brexit."
She added: "There is a danger that we will start finding ourselves, or the Government will find itself, in a position where other people are setting the terms of the debate."
She took an apparent swipe at Brexit Secretary David Davis, who has suggested Britain would quit the single market to gain control over its borders.
Mrs Morgan said there are "senior ministers who are keen on a hard Brexit" who have spoken out against any freedom of movement and warned "the Prime Minister has to be the one leading this, and I don't think that is something she would want to see".
Mrs Morgan said Mrs May should give a "broad outline" of her Brexit negotiating position.
"There does need to be a clear plan from the top of Government about what it is that we are looking for," she said.
Asked when this should happen, Mrs Morgan said: "Certainly in the next couple of months.
"There is obviously an opportunity - the Prime Minister will be giving her speech at the party conference next week."
She said Mrs May does not need to give "all the details" of her negotiating stance but should give a "broad outline" including her position on the balance between the freedom of movement and access to the single market.
Meanwhile, a German business boss has urged Britain to go for a hard Brexit, saying this would be better than a "fudge".
Markus Kerber, head of the BDI, which represents German industry, told Today: "We have a rough idea of what the British Government wants to see - it wants to have a relatively full access to the single market and yet limited or non existing freedom of movement of labour charter.
"That, I think, is impossible at the moment.
"So what we think the British Government wants, I can tell you straight away, it's not what the Continental Europeans are willing or even able to give."
He said freedom of movement cannot be hived off from the other pillars of the single market, insisting "for us the single market and Eastern Europe and the freedom of movement are one deal that is inseparable".
He added: "I do agree with those commentators who would say it is better to have a hard Brexit that works than to have a fudge in the middle that may have to be renegotiated or doesn't politically work, and you have uncertainty lingering on.
"If British decision-makers look very hard at what it is that they want and what it will be that they get, there is no other option than the hard exit."