That stance suggests a relationship far closer to the status quo, which largely suits Brussels, than EU membership.
Unless, of course, Labour caves completely to the EU’s demands, which is what Brexiteers must fear.
Mr Barnier is an influential pundit on matters like Brexit but his sway should not be overstated.
Under the French system, it is the president, Emmanuel Macron, who is responsible for foreign policy and not the prime minister.
He always drove for the toughest line on Britain during the negotiations anyway to sap support for the Frexit-supporting Marine Le Pen.
Brussels showered warm words on Mr Barnier, a commissioner for 15 years and a member of the centre-Right Republicans, after his elevation.
But he does not get on with Ursula von der Leyen, the European Commission president. He refused to back her nomination for a second term and has muttered darkly about German domination of the EU.
Her interlocutor in France will be Mr Macron, who Sir Keir has courted assiduously and who played a crucial role in appointing Mrs von der Leyen in her first term.
The commission, which handles Brexit talks, is unlikely to pink its red lines for Labour anymore than it did for the Tories.
However, the anglophile Mrs von der Leyen, a former LSE student, is more likely to back a last-minute fudge than Mr Macron or Mr Barnier.