"Ninety percent of the accord on the table has been agreed with Britain", he told France radio.
"We are working to ensure that we have that future relationship in place by the end of December 2020", May said as she arrived at European Union headquarters in Brussels on Thursday for meetings on migration, security and other issues.
Referring to the Irish border question, he said: "From the beginning of the mandate that was given to me, we always said that it was an issue that needed to be resolved".
British Prime Minister Theresa May says the remaining differences on Brexit terms between the United Kingdom and the European Union are "few but considerable" - but she says she is confident the two sides will reach an agreement.
To try to move the talks forward, May had earlier signalled she would consider extending a so-called transition period "for a matter of months" after Britain leaves the European Union in March, a move her critics called a betrayal but one which the bloc welcomed.
The mooted special November summit to sign a withdrawal agreement may not be in the diary, but neither is the threatened "no-deal" November summit.
The Foreign Secretary also called on his party to unite behind the Prime Minister as the negotiations enter the final phase.
"I am open to the idea of an extension or a longer transition period, but that's not an alternative to a legally binding Irish backstop, so it's certainly not an alternative; it's something that might be part of the mix", he said. Her strategy, therefore "seems to be to leave everything until the very last minute... and hope she can cobble together a deal before bouncing parliament into supporting it for fear of something worse".
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But the Democratic Unionist Party (DUP), which props up May's government, says it will withdraw its support if this happens. What's even worse is she is now going to put us in a position that, six years after voting to leave the European Union, we will still be in the EU.
The meeting in Brussels was supposed to be the occasion when the leaders of the European Union 27 member states gave the green light for a special summit in November to finalise the terms of Britain's withdrawal.
British Prime Minister Theresa May also spoke about "working intensively over the next days and weeks" to achieve agreement that avoids a no-deal departure from the bloc on March 29 that could create chaos at the borders and in the economy.
"We are in a negotiation but at the moment it begins to look more like a capitulation than a negotiation".
Simon Collins, the Executive Officer of the Shetland Fishermen's Association, warned the Government it should not let fishermen down again - insisting the key issue between the fishing industry and the UK Government remains to be "trust" in Brexit talks.
Her critics have the numbers to do this but say they have held off so far because they believe she would survive.
The problem centers on a so-called backstop - an insurance policy to ensure there will be no return to a hard border on the island of Ireland, a former focal point for sectarian tensions, if a future trading relationship is not in place in time.
For all her "wittering" at the dispatch box about backstops, she "can get nothing worth having through parliament or the EU".
Both sides agree there must be no hard border, but each has rejected the other side's solution. Under the current proposal, the United Kingdom would effectively be a non-member: it would remain part of the single market and customs union, but would no longer be represented in EU institutions.
EU Brexit negotiator Michel Barnier sought to calm the mood, saying the only remedy was to use the remaining period between now and Christmas.