This follows fresh reports that Northern Ireland could retain “regulatory alignment” with the EU after the UK exits the EU in 2019.
European Council President Donald Tusk also said he was “encouraged by progress” and a deal on Ireland, the divorce bill and citizens’ rights was “getting closer”.
UK Prime Minister Theresa May was meeting EU figures, including Commission President Jean-Claude Juncker, in Brussels on Monday in an attempt to finalise the deal ahead of a summit in 10 days’ time.
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Speaking on the talks, Lamberts, a member of Parliament’s Brexit steering group, said, “It seems the British government is now coming to terms with reality and is finally willing to make the necessary concessions to allow us to move on to stage two of the negotiations.
“While the hard-line Brexiteers will not be happy, the necessity of a special agreement for Northern Ireland has been clear to all rational observers since day one.”
The Belgian MEP added, “Maintaining regulatory alignment between Northern Ireland and the Republic of Ireland is the only solution if the Good Friday agreement is to be respected. I am optimistic that the European Council can now agree to move discussion on to the UK’s future relationship with the European Union.”
Lamberts met with EU Brexit negotiator Michel Barnier and Juncker on Monday morning.
“It seems the British government is now coming to terms with reality and is finally willing to make the necessary concessions to allow us to move on to stage two of the negotiations" - Philippe Lamberts
MEPs were allegedly told by Barnier that Theresa May had conceded after days of intense talks that the province would be treated as a special case.
A draft of the text of a 15-page joint statement between the Commission and the British government is said to include a commitment in paragraph 48 that “in the absence of agreed solutions the UK will ensure that there continues to be continued regulatory alignment” with the internal market and customs union.
However, elsewhere there was a mixed reacted to reports that Northern Ireland could retain "regulatory alignment" with the EU.
Scottish First Minister Nicola Sturgeon said there was “surely no good practical reason” why other parts of the UK could not do the same.
She said, “If one part of UK can retain regulatory alignment with EU and effectively stay in the single market (which is the right solution for Northern Ireland) there is surely no good practical reason why others can’t.”
Irish Taoiseach Leo Varadkar was expected to make a statement later on Monday.
DUP leader Arlene Foster has said her party “will not accept any form of regulatory divergence” that separates Northern Ireland from the rest of the UK.
She went on to accuse the Republic of Ireland of “seeking to unilaterally change” the Good Friday agreement without the DUP’s “input or our consent”.
London mayor Sadiq Khan said, “It will have huge ramifications for London if Theresa May has conceded that it’s possible for part of the UK to remain within the single market and customs union after Brexit. Londoners overwhelmingly voted to remain in the EU and a similar deal here could protect tens of thousands of jobs.”
Speaking on Sunday, Simon Coveney, Ireland’s deputy Prime Minister, said his country has “no desire” to delay the UK’s Brexit talks.
However, the former MEP also warned that not enough progress had been made on the border with Northern Ireland.
While Ireland was not looking for “full detail”, it was seeking the parameters within which a solution could be found.
Brussels said it will only recommend the start of talks about future trade arrangements when it deems sufficient progress has been made on three issues - the status of expat citizens, the divorce bill and the Northern Ireland border.
The UK has been set a deadline of this week to come forward with an improved offer on them, and hopes that the go ahead for future talks will then be given at an EU leaders’ summit on 14-15 December.
On the divorce bill, the UK is understood to have recently increased its offer, which could be worth up to €50bn (£44bn).
On the issue of rights for the three million EU citizens in Britain, the UK has agreed that those who already have permanent residence will not have to pay to apply for settled status. Those making a first-time application for the right to stay after Brexit, however, will face a charge - reportedly similar to the cost of applying for a passport.