London has demanded an opt-out from the "ever closer union" principle enshrined in the EU's treaties as part of reforms it wants to agree before holding a referendum on its membership of the bloc.
At informal talks in Rome, the foreign ministers of Belgium, France, Germany, Italy, Luxembourg and the Netherlands issued a joint communique in which they admitted to being "concerned about the state of the European project".
They said the EU was facing "very challenging times" due to the migration crisis and the threat posed by terrorism.
And they insisted that, for them, the answer lay in more, not less integration while also acknowledging that not every country in what is now a 28-member bloc should have to agree.
"We firmly believe that the European Union remains the best answer we have for today's challenges and allows for different paths of integration," the communique read.
"We remain resolved to continue the process of creating an ever closer union among the people of Europe."
Tuesday's dinner meeting was called by Italy, whose centre-left government has made it clear it wants a core of EU countries to forge ahead with steps towards further integration, with moves towards a banking union, tighter fiscal harmonization and increased political and security cooperation the areas where they see change as most desirable.
Rome has also indicated that it is relaxed with countries like Britain limiting their engagement with the EU to essentially being part of a large free trade area - an option Italian officials say they would much prefer to the unpredictable scenario of a "Brexit", as Britain's possible departure has been dubbed.
Brexit referendum looms
British Prime Minister David Cameron is hoping to tie down a package of reforms at a summit of EU leaders on February 18-19th.
If he succeeds, he is expected to move quickly to hold an in-out referendum in which he will argue that membership no longer poses a threat of further erosion of British sovereignty.
A two-speed EU is already a reality to an extent, with only 19 of the 28 member states having adopted the euro single currency - although all the other countries, except Britain and Denmark, are theoretically committed to working towards joining.
But the idea of formally enshrining the "two-speed" principle has long been taboo among supporters of deeper integration who often argue that the Union is like a bicycle which can only balance when it is moving forward.
Italy meanwhile will ask its citizens how they want to see the EU develop via an online consultation to be launched on February 12th, parliamentary speaker Laura Boldrini announced earlier on Tuesday.
The six-question survey will seek to establish what Italians see as the EU's strengths and weaknesses, the areas in which it could do more, how it could be more efficient and how to reform its institutions.
"We are facing major challenges - climate change, terrorism, migration - and no state can face them alone," Boldrini said in comments which reflect the view of much of the Italian political class but not necessarily those of voters.
"United we can influence events, divided we become totally insignificant."