Barnier's statement has come before a government Brexit meeting, called by the British Prime Minister on May 15.
"Nobody is saying that everything has to be resolved by the end of June but I think we need to see a solution on the Irish Border backstop issue taking shape by the end of June", he said.
The UK rejected a subsequent attempt by the European Union to translate that agreement into legally operable text in a proposed withdrawal treaty.
According to Reuters, diplomats and EU officials noted the lack of progress in negotiations on Brexit with London since the last meeting of EU leaders, voicing doubts about the fact that Britain and the EU will be able to reach important decisions in this matter to the next high-level summit which will be held on 28-29 June 2018.
Ireland and the Commission have been demonstrating a new willingness to explore what is reported to be the British prime minister's preferred option, a customs partnership, of which they are sceptical.
"I just cannot see how the United Kingdom being outside the customs union can possibly work with an open border on the island of Ireland". "I very much hope we will agree but there are no guarantees, unfortunately".
Simpson wins PLAYERS by four strokes, Woods signs off 11th
He bogeyed the hole on Saturday. "I told him, 'Pretty inconsistent.' He asked me had I ever tried the claw grip, and I said no". The 14th hole had been somewhat of a nemesis for Woods this week, and it did him no favours on Sunday.
Mr Barnier added: "The clock is ticking and we have to conclude and to finalise this agreement. before October or November, because we have to keep the time for the ratification by the European Parliament, the European Council and the British side".
"I think the concept of a shared customs space is something that we would like to see some new thinking on from the British side", he said before holding Brexit talks with other European Union ministers in Brussels.
Carwyn Jones said he cannot see how "an open border on the island of Ireland" can be maintained without some form of a UK-EU customs union. It would make a lot of other problems that are hard to solve right now...
It is not clear how the ideas of "shared customs space" and "customs partnership differ precisely except that the name of the former may be seen as more politically palatable to Brexiters".
"October is only five months from away, but and some key issues related to the withdrawal agreement need to be settled".
Mr Barnier said there had been "misunderstandings" in the coverage of the story, adding: "We are not kicking the United Kingdom out of Galileo".