Barnier, however, is said to have rejected this idea out of hand, saying that such a scheme would not allow the European Union to unilaterally make decisions about equivalence, something it wishes to do.
Mr Raab, however, says he is still hopeful a deal can be concluded this year.
Having only taken the position less than three weeks ago, Mr Raab put a fearless face on and told his negotiating partner: "We have work to do".
After a chaotic two weeks that have seen senior ministers quit her government and rebellions from both wings of the ruling party, premier Theresa May faces further pressure after a poll for The Sunday Times showed that the public was also deeply opposed to her strategy.
Replying to those comments, Lineker tweeted he was "happy to admit I've now changed my mind", adding that he did not envisage "at the time that we'd ever contemplate a "no deal Brexit" and the ensuing carnage that would bring".
"The EU can not and the EU will not delegate the application of its customs policy, of its rules, Value-Added Tax and excise duty collections to a non-member who would not be subject to the EU's governance structures", he said. But he had added: "On our future economic relationship it comes as no surprise that finding common ground between the EU27 and the United Kingdom is more hard".
The ex-footballer has joined forces with the People's Vote group as they launch a series of rallies and protests across the United Kingdom calling for a referendum on the final Brexit deal.
Mr Raab acknowledged that the Chequers plan struck a balance between the UK's ability to diverge from the European Union and the need to protect trade and address the Irish border issue.
It also points out that it is already preparing to relocate or reassign tasks that are now performed in the United Kingdom - such as the Galileo Security Monitoring Centre (part of the EU's project for satellites in space) and the UK-based EU Reference Laboratories - because it will not be possible "to entrust a third country" with such EU tasks after the withdrawal date. They hope the United Kingdom government will gradually shift its red lines and opt to remain closer to the bloc.
"We have no objection in principle to this", Barnier said.
"Exiting the EU without agreement on this matter would leave regulators in limbo in March 2019, and even under the terms of the draft EU-UK withdrawal agreement, access to such databases would be switched off post-2020", it suggested.
Previously, May and Chancellor Philip Hammond had favoured mutual recognition - which would have meant the United Kingdom recognising European Union regulations around the financial services industry, and the European Union doing the same with the United Kingdom, with both sets of rules closely aligned.