Michel Barnier imposed the deadline in a move likely to heap more pressure on Theresa May over progress of the discussions.
The PM said previously she hoped and had a “degree of confidence” that phase-two Brexit talks could start before Christmas.
But speaking at a news conference after the sixth round of negotiations ended in Brussels, Mr Barnier issued a new warning.
Asked to confirm reports that the EU needed clarifications or more concessions from the UK in the next two weeks to move discussions on next month, he gave a one word answer: “Yes.”
Mr Barnier said only “real and sincere progress” on the sticking points of citizens’ rights, the Irish border and exit bill would be enough.
Failing that, he warned, negotiations on Britain’s future relationship with the EU “will be put back”.
In a news conference light on new announcements, Mr Davis called for Brussels to show “flexibility, creativity and imagination”.
He said the UK was “ready and willing to engage as often and as quickly as is needed to secure this outcome over the weeks remaining ahead of the December European Council”.
The Council meeting where EU27 leaders will decide if “sufficient progress” has been made begins on 14 December.
Some in the Conservative party will ask why the two-week deadline has been imposed with the summit still over four weeks off.
Tory MP Iain Duncan Smith suggested giving Mr Barnier a reciprocal ultimatum. He told Sky News: “We should tell him he has two weeks to sort out an FTA (Free Trade Agreement).”
Sky’s Tamara Cohen said: “A one-word answer from Michel Barnier – ‘yes -‘ has set the two-week timer ticking on the Brexit divorce bill.
“The reaction from Brexiteers in the Conservative to the idea of paying billions to Brussels in a divorce settlement is more muted than it was a year ago when the Government was riding high.
“Theresa May is their biggest hope of delivering Brexit and they know it would be detrimental to their interests to undermine her, amid all her other woes.
“Ahead of the 14 December summit, she’ll need to sound out what they’ll accept. And if Britain is to up its offer, what concessions may be on the table for trade or the transition period.”