The former British foreign secretary's warning came as Mrs May gave one of her most positive forecasts yet for Britain's future after Brexit, telling the BBC: "I believe that our best days are ahead of us".
She agreed past year for Northern Ireland to stay locked in the bloc's Single Market if the border can not be kept open.
The International Monetary Fund (IMF) has warned Britain that it should expect "substantial costs" to its economy if it leaves the European Union without agreeing to transitional arrangements.
Although she is signed up to the principle of a backstop if no better arrangement can be agreed, Mrs May has robustly rejected the version put forward by Brussels, which would see Northern Ireland remain within the European Union customs area and effectively draw a customs border down the Irish Sea.
Ms Lagarde said that the United Kingdom would need to negotiate 63 trade deals in the event of a "no-deal" departure from the European Union.
But even if she gets an accord in the coming weeks, it must be signed off on in parliament, where she can only muster a slender majority.
It showed that most trade between the United Kingdom mainland and Northern Ireland was everyday movement of goods for high-street stores rather than major shipments of produce carried out by large companies.
But many business chiefs and investors fear politics could scupper a deal, thrusting both the European Union and the United Kingdom into a "no-deal" Brexit that they say would weaken the West, panic financial markets and block the arteries of trade.
Mrs May also warned that the only alternative to Parliament passing her proposals would be no-deal.
May also insisted no other plan on the table would ensure "frictionless" trade in Ireland.
That would see goods tracked using a barcode and undergo customs checks on ferries or on arrival at factories, according to a separate report in the Financial Times.
The problem she faces is that any agreement based on keeping close trading ties to the bloc puts her on a collision course with hard-line eurosceptics in her Conservative Party - and could trigger a leadership challenge by Mr Boris Johnson.
"The whole thing is a constitutional abomination, and if Chequers were adopted it would mean that for the first time since 1066 our leaders were deliberately acquiescing in foreign rule", Johnson said, referring to the 11th-century invasion which established Norman rule over England.
"It would inevitably have a series of consequences in terms of reduced growth going forward; increased deficit, most likely. and depreciation of the currency", said Lagarde, the IMF's managing director.
French President Emmanuel Macron (R) chats with Austrian Chancellor Sebastian Kurz after a meeting at the Elysee Palace in Paris on September 17, 2018.
"But I can say at this point, without disclosing any numbers, is that clouds on the horizon have not become lighter but darker", she said.