The sterling which had traded as high as 1.3072 against the US Dollar slipped to 1.2969 at the NY close and to 0.8742 versus the euro having been as high as 0.8692, just two points above Thursday's six month sterling top of 0.8682.
The brother of former foreign secretary Boris said: "It has become increasingly clear to me that the withdrawal agreement, which is being finalised in Brussels and Whitehall even as I write, will be a awful mistake".
May is expected to call a special cabinet meeting next week as she tries to persuade ministers to unite behind her proposals for the Irish backstop.
Such a choice was a "failure of British statecraft on a scale unseen since the Suez crisis", he added.
May's Downing Street office said on Twitter: "We will not under any circumstances have a second referendum".
Boris Johnson was one of the most prominent and influential of the Brexit advocates.
Britain has proposed a UK-wide temporary customs arrangement with the European Union to resolve the issue but Brexiteers in her party want London to have the final say on when that arrangement would end, to prevent it from being tied indefinitely to the bloc.
Leading Tory Brexiter Jacob Rees-Mogg rejected calls for a second referendum, but told the BBC he agreed with criticism of the deal.
In a tweet, Boris said he had "boundless admiration" for his brother and that they are "united in dismay" at the indefensibility of May's position even though they disagree over Brexit.
In contrast to his brother, however, Mr Johnson voted Remain in the European Union referendum, and is now pushing for a second vote on leaving the bloc.
However, there are concerns that a second referendum could embolden the far-right in the United Kingdom, while Eurosceptics point to examples where votes within EU member states rejecting greater federalism and austerity have been overturned in what they say is yet more evidence that there is a "democrat deficit" within the union.
The remain-supporting brother of Boris Johnson, whose bombshell resignation took Westminster by surprise on Friday afternoon, said if other senior Conservatives followed suit, "good on them".
Johnson said this morning that other ministers are thinking hard about the Brexit and its implications.
"In fact so great is the gulf now between what was promised in the referendum campaign and what is now on offer in the Prime Minister's proposed deal that I have had no choice but to submit my resignation to the Prime Minister this afternoon".