The UK formally left the European Union back in January after voting to leave in 2016 and negotiations are currently deadlocked with both sides unable to come to an agreement.
As the end of the transition period draws closer and closer, Prime Minister Boris Johnson has vowed to end the year with or without a trade deal in place.
As Mr Frost is set to take up a new post as National Security Advisor (NSA) in September, the chief Brexit negotiator will continue to oversee talks with the EU.
He is believed to have told colleagues he will stay on if a trade deal has not been agreed with Brussels by autumn.
Mr Frosts appointment as the NSA is believed to be a signal to the bloc that Britain will walk away if a deal is not struck over the summer.
A government source told the Sunday Telegraph: Hes said he will stay in charge of the negotiations until they have been completed.
He will take up the new post in September by plans to spend 90 percent of his time on the trade talks if thats what is needed.
When his new role was announced Mr Frost said EU talks would remain my top single priority until those negotiations have concluded, one way or another.
The next round of Brexit talks are due to resume on August 17.
READ MORE:Brexit warning: Frost told EU not fazed by no deal threats
Following the announcement of his new role, former Prime Minister Theresa May condemned the move by Mr Johnson.
She said: Why is the new national security adviser a political appointee with no proven expertise in national security?
Cabinet Office minister Michael Gove defended the move by the Prime Minister and said Mr Frost was highly qualified and would be accountable to Mr Johnson.
Mr Gove said the negotiator would neither be a civil servant nor a special adviser but have the status of an envoy.
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Mr Johnson called Mr Frost as experienced diplomat, policy thinker and proven negotiator, with a strong belief in building Britains place in the world.
He added: In his new role I am confident he will make an equal difference to this countrys ability to project influence for the better.
As well as his new role, Mr Frost was also given a peerage by Mr Johnson after he was one of his key advisers as foreign secretary.
The deadline for negotiations has been pushed back to October following the ongoing coronavirus pandemic but there is no sign of an agreement yet.
One of the main disputes in the negotiations is fishing as the EU wants to maintain its rights to Britain's fishing waters.
Under the controversial Commons Fisheries Policy (CFP), all member states are given access to EU waters via quotas.
As the UK has a large coastal area, critics have often argued the system is unfair.
Back in June, the EUs chief Brexit negotiator Michel Barnier suggested the UK would be treated as an independent coastal state.
According to reports at the time, Mr Barnier told other diplomats a compromise would have to wait until other parts of the trade deal were being finalised.
EU sources said the Frenchman is looking for a whole trade deal which relies heavily on fishing rights.
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