As Scottish First Minister Nicola Sturgeon began her diplomatic mission to Brussels in an attempt to maintain Scotland's place in the EU, Rebecca Harms, Co-Chair of Parliament's Greens/EFA group, pledged its support.
She said, "We will support exploring all the options that would allow pro-European Scotland to remain within the EU, as they have cleared voted do."
She was speaking after meeting Sturgeon in Parliament in Brussels on Wednesday, one of a series of meetings planned between group Chairs and the SNP leader.
After his meeting with Sturgeon, Gianni Pittella, leader of Parliament's S&D group, told this website, "We just expressed our sadness for the unfortunate outcome of the referendum and at the same time the sympathy to the Scottish attachment to the EU integration."
The Italian deputy said, "We reassured Sturgeon we'll closely follow how this new situation will internally develop in the UK."
Sturgeon has already said she would tell her MSPs to refuse "legislative consent" if and when the Scottish Parliament was required to ratify the UK's withdrawal from the EU.
Polls reported by two Scottish newspapers at the weekend suggest she might conceivably win a majority among the Scots for secession.
The Glasgow-based Daily Record carried a post-Brexit poll on Saturday that found 54 per cent of Scots voters would vote yes to independence.
The following day the Dundee-based Sunday Post highlighted a survey conducted on its behalf had found 59 per cent in favour of independence.
Although Sturgeon gained cross-party support at Holyrood for her mission to Brussels on Wednesday, she noticeably failed to get a meeting with Donald Tusk, President of the European Council. Instead, she had a 30 minute meeting with Parliament's President, Martin Schulz, before seeing group leaders.
Meanwhile, Irish MEP Martina Anderson says that while she is critical of the EU's leadership, she insists that the majority who voted Remain in the North of Ireland must "not be ignored" in the upcoming EU exit negotiations.
In a debate on the issue in Parliament on Tuesday, she said, "We ask you to respect our vote. We stand by the vote of the people of North of Ireland, which, just like Scotland, voted remain."
"We accept, we respect and we will defend the wishes of the people of the North of Ireland. However, if English votes drag us out of the EU, that would be like Britannia waives the rules."
"There was a democratic vote - we voted to Remain. I tell you that the last thing the people of Ireland needs is an EU border with 27 member states stop right in the middle of it," she told the packed session.
At present, there is not a majority for Britain to leave the EU in either the House of Commons or the House of Lords. Given a free vote, there are estimates that the unelected Lords would probably reject Brexit by a margin of six to one.
One EU insider said that the Commons might, for instance, seek to prevent the UK Prime Minister from triggering article 50, the clause of the Lisbon treaty that provides the framework of an EU member state's exit from the Union
There is also pressure to hold a second referendum. Few UK politicians - fearful of challenging the verdict of an already angry electorate - are likely to articulate such an argument in public.
But Jeremy Hunt, the UK Health Secretary, has made the case for a second referendum or another general election on the negotiated terms of exit.
Robin Butler, the former head of the UK civil service, has suggested the same.
On a visit to Brussels on Tuesday, the US Secretary of State, John Kerry, raised doubts about whether Brexit will ever happen, suggesting most Leave campaigners do not truly believe in Britain's divorce from the EU and do not know how to achieve it.
Claiming there were a number of ways in which last Thursday's vote could be "walked back", Kerry, who visited Downing Street on Monday, said Cameron was loath to invoke article 50, the EU exit procedure.
He said Cameron felt powerless to "start negotiating a thing that he doesn't believe in" and "has no idea how he would do it."
Meanwhile, billionaire George Soros, the man who predicted that a UK vote to leave the EU would bring about "Black Friday", has said that UK's surprise decision to leave makes an eventual dissolution of the EU "practically irreversible".
His comments come on the eve of a debate he is due to take part in at the European Parliament on Thursday.
"Now the catastrophic scenario that many feared has materialised, making the disintegration of the EU practically irreversible," Soros, founder and Chair of the Open Society Foundation, said.
The billionaire went as far as saying the country itself "may not survive" the decision, alluding to the possibility Scotland, which voted overwhelmingly to remain in the EU, will gain independence.
On Thursday, Soros will join MEPs in discussing the budgetary implications of the refugee crisis.
Others include the directors of Frontex, the EU Asylum Support Office and UNHCR Europe, Commission representatives and the Secretary-General of the European Council on Refugees and Exiles. The hearing is organised jointly by the budgets, civil liberties, foreign affairs and development committees.