The next round of talks is due at the end of the month, with both sides looking for progress towards a solution to three of Brexit's thorniest problems: how much Britain should pay to leave, what rights British and European Union citizens will have, and how to manage a land border to the bloc in Ireland.
The two leading politicians said the government wanted to ensure "there will not be a cliff-edge when we leave the EU".
"We've been crystal clear that issues around our withdrawal and our future partnership are inextricably linked, and the negotiations so far have reinforced that view".
This week we're expecting to find out the UK Government's position on the border. The EU has warned that an already-tight timetable could be delayed ahead of a scheduled March 2019 exit.
In the coming week the British government is to begin publishing papers that will have more details of its policies and goals in relation to the Brexit process.
Theresa May is to offer free movement to Irish citizens in and out of Britain after the United Kingdom leaves the European Union, the Sunday Telegraph can reveal. The Brexit department declined to comment on the story.
Officials said the papers would show the British government was ready to "broaden out" the negotiations and move forward towards a deal that worked for both sides.
Weekend tropical outlook: A depression could develop in the Atlantic
It is not expected to develop into a hurricane, as conditions will not be conducive for such development, AccuWeather reports. At 5 a.m., the center of Tropical Depression No. 8 was located about 310 miles north-northeast of the southern Bahamas.
That will come alongside other documents setting out how Britain's involvement in the customs union with other EU countries this week.
The jointly penned article appeared created to reassure those in favour of a clean break, but will do little to comfort those fearing the consequences of a so-called "hard Brexit".
Also, details of future arrangements including Britain's proposals for a customs agreement with the EU will be released prior to a meeting of the European Council in Brussels in October.
The first will be a proposal for new customs arrangements.
Barnier has maintained that negotiators must make progress on the rights of European Union and British citizens, the border with Ireland and Britain's exit payment before discussing a trade deal, while Prime Minister Theresa May wants an accord before leaving.
Writing for the Daily Telegraph, the remain-backing Finance Minister Philip Hammond and the Eurosceptic Trade Minister Liam Fox said Britain would leave the European Union in March 2019, there would be a short "interim period" to smooth the transition during which Britain would not be party to European Union treaties and after that Britain would become fully independent.
Some still dispute her approach, with former foreign minister for the now-opposition Labour party David Miliband and pro-EU Conservative MP Anna Soubry warning in Sunday newspapers of the economic "self-harm" of leaving the single market.
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