The comments from commission president Jean-Claude Juncker's right-hand man heap further pressure on the Prime Minister as she prepares to go to Brussels to seek a renegotiation of the UK's Withdrawal Agreement.
May, during a speech on Tuesday in Belfast, restated her "unshakeable" commitment to avoiding a hard border and said she didn't plan to remove the "insurance policy" entirely.
Downing Street has indicated potential solutions could revolve around a time limit or unilateral break clause on the backstop or new technologies to make it unnecessary.
Mrs May told reporters she had asked Mr Javid and the Northern Ireland secretary of state Karen Bradley to look at ways of securing the rights of Irish citizens living in Northern Ireland post-Brexit to bring in family members from other non-UK countries to take up residency as "a matter of urgency".
Meanwhile, former Northern Ireland first minister Lord Trimble is threatening legal action over the Irish backstop, which will prevent the return of a hard border.
"The Prime Minister has effectively run down the clock and therefore it is impossible to see a way forward without a backstop".
As Theresa May travels to Brussels to try and win concessions on the Irish border backstop, European Union officials are already discussing awarding compensation to Ireland in order to prepare for the growing fear of the United Kingdom crashing out of the bloc without a deal.
British ministers, The Sun newspaper said, have been examining a plan drawn up by Japan's Fujitsu to track trade across the border, while the Telegraph said ministers had discussed delaying Brexit by eight weeks.
She has assembled a taskforce of rebel MPs to help her come up with "alternative arrangements" to the controversial backstop.
Mrs May's spokesman insisted: "She is absolutely determined to deliver Brexit on time".
She says "one has to be creative, and we must listen to one another" but that an agreement on the Irish border is still possible.
Gloria Guevara, President & CEO, WTTC, said "a "no deal" Brexit would have a dramatic impact on one of the U.K.'s most significant sectors".
She told a meeting of businessmen and women in Belfast that there were "a number of ways" to amend the amend the backstop, which the Biritsh parliament has demanded before agreeing a divorce deal with Europe.
European Union summit chairman Donald Tusk said he had abandoned hope that Brexit might be stopped and said on Wednesday his priority was now to avert a "fiasco" in 50 days if Britain crashes out without a deal.
"It's very clear from our perspective that the time for reiterating red lines and regurgitating reassurances has long gone", she said. "If British politics can not accept the specific needs of this country, of the north of Ireland, if British politics is incapable of acknowledging and upholding the Good Friday Agreement and preventing a hardening of the border, then the only last option - the backstop of last resort - is a referendum on Irish unity", McDonald said.
The free flow of people and goods across the near-invisible frontier now underpins both the local economy and Northern Ireland's peace process.
Firstly, the European Union is currently refusing, for now, to renegotiate the Brexit withdrawal deal.
Without the protections offered by the backstop, future regulatory divergence between the United Kingdom and the Republic of Ireland, in areas such as agriculture, could necessitate the return of customs checks and therefore a "hard border".