The current limbo for Brexit is set to last until early next week after EU ambassadors failed to reach an agreement over the U.K.'s request for a deadline extension.

On Friday morning, EU ambassadors discussed what sort of delay should be granted to the U.K. for its current Brexit deadline of October 31. They accepted that a delay was needed but couldn't agree on a firm date.

"There was full agreement on the need for an extension. There was full agreement to reach a unanimous, consensual EU27 decision and there was full agreement to aim to take the decision by written procedure," an EU source, who did not want to be named due to the sensitivity of the talks, told CNBC Friday about the ambassadors meeting.

"Work will continue over the weekend," the same source told CNBC, adding that the ambassadors are set to meet again "early next week" to finalize the agreement.

The ambassadors to the EU present their country's position and prepare the background work for European leaders, who make the final call on most policies, including granting the U.K. a third Brexit extension.

France, which has taken the toughest position on Brexit deadlines compared to the other EU countries, does not want the U.K.'s departure to interfere with the EU's upcoming political cycle.

The European Commission — the executive arm of the EU — will see a new team installed in the coming months. Granting a Brexit delay beyond this start date would mean these new officials would have to deal with the U.K.'s departure and potentially be forced to include a representative from the U.K. too. Some believe the next Commission should be focused on other policy areas rather than Brexit.

Amélie de Montchalin, the French minister for European affairs, told RTL radio on Wednesday that France is not seeking to give the U.K. an ultimatum. "The question is knowing why we should give more time. Giving more time alone is not a solution," she said.

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Earlier this week, she made the point that the EU renegotiated the controversial Irish backstop, a major sticking point in discussions, with U.K. leader Johnson and it was now up the Britain to say yes or no to the revised agreement.

A second EU official, who did not want to be named due to the sensitive nature of the discussions, told CNBC earlier on Friday that "France wants to engineer a quicker exit."