The former Attorney General labelled the Prime Minister’s deal proposed to the European Union as looking “pretty second rate”.
The Tory MP was among an estimated 700,000 people taking part in the People’s Vote march through London on Saturday, in what had been billed as the largest anti-Brexit demonstration too date.
Mr Grieve told BBC Radio 4’s Today programme: “If we don’t do this now then we will lose the opportunity and once we are out of the EU on March 29, going back in is a completely different thing to deciding to stay in.
“My view has grown over the last few months that in fact we are reaching a dead end and that there may be some sort of deal on offer but one that i have to say looks to me pretty second rate.
“When you get this sort of problem, in view of the first referendum, the only solution is to go back to the public and say is this what you really want. And if they do want it, so be it.”
The Remainer MP claimed the EU referendum in June 2016 was based on “abstract discussion” and warned Britain is on course to deliver a Brexit that “very few people want”.
He added: “I think this is the end. If people find that what they are about to get is what they want then I as a democrat would accept it.
“But what worries me is I think we are about to deliver a Brexit that I think, in fact, very few people want.”
Following yet another stalemate in talks with EU leaders in Brussels, Theresa May said she would consider extending the transition period beyond December 2020 to allow the UK and EU to rubber-stamp problems around the future relationship post-Brexit, most notably, the issue of the Irish border.
But this move has sparked a backlash from Tory MPs, with arch-Brexiteer Jacob Rees-Mogg dismissing it as “a rather poor attempt at kicking the can down the road”.
Mr Grieve conceded it was likely the transition period would be “extended well beyond three years as we try to thrash out the continuing relationship”.
He added: “That highlights for me why in fact we are embarked on the project which is totally different from what people might have had any reason to expect in 2016.
“We are in the most extraordinary situation where we remain tied to all of the rules of the EU without any ability to influence the input.”
On Friday, Jeremy Hunt urged warring Tories to get behind Mrs May’s Brexit strategy as she continues to face a fierce backlash against the possibility of extending the transition period.
The Foreign Secretary said an EU proposal to continue the implementation process “might help” get an agreement on the future relationship and rejected claims that it amounted to a “capitulation”.
Speaking on BBC Radio 4’s Today programme, he said: “The reason why this week has been difficult is because Theresa May has not buckled.
“She has held firm. She has stuck to her principles.
“It is precisely because she has not capitulated that we have not concluded this agreement.
“The one thing I would say to my colleagues is that the great strength of the EU in these negotiations is that the 27 EU nations have remained united.
“We now need to do the same behind Theresa May to maximise her negotiating leverage in Brussels and make sure she does come back with that deal that honours the letter and spirit of the referendum decision.”
But the EU’s chief negotiator Michel Barnier warned the UK’s withdrawal agreement could still fail over the issue of the Irish border, leading to an “extremely serious” no-deal Brexit.
Speaking to France-Inter radio, he said the deal was 90 percent” complete and was hoping to complete it “in a few weeks or a few months, as soon as possible”.
But asked if he was convinced an agreement would be reached, he replied: “I have no deep conviction on this subject, because in the UK the political situation is very complex and I don’t know what decisions Theresa May will take.”