(Updates with EU official)
By Gabriela Baczynska and John Chalmers
BRUSSELS, Oct 19 (Reuters) - European Union chiefs are so fed up of Brexit that when they clinched a new divorce deal with Prime Minister Boris Johnson this week they did not want to entertain the need to delay Britain's departure beyond Oct. 31.
But once Britain sends a letter to seek an extension for the divorce - as now seems likely after Johnson's failure to get the agreement approved in parliament on Saturday - there is very little chance that the other 27 members states would refuse.
After a call between Johnson and European Council President Donald Tusk on Saturday evening, an EU official made clear, however, that the bloc would not rush.
"The prime minister confirmed that the letter would be sent to Tusk today," the official said. "Tusk will on that basis start consulting EU leaders on how to react. This may take a few days."
An EU diplomat reinforced that message: "We won't be rushed by any request."
The bloc hopes the deal can still pass in the factious House of Commons in time to let Britain leave with a deal on Oct. 31.
The EU would, however, need to step in should that start looking unlikely if it wants to avoid a no-deal Brexit at the end of the month.
"I can't see them refusing," Anand Menon, director of UK in a Changing Europe and a Brexit expert, told Reuters. "They don't want no deal and they certainly don't want to be blamed for it."
EU leaders might end up agreeing any new Brexit date at a hastily convened emergency summit, possibly next weekend. Their ambassadors are due to meet in Brussels on Sunday morning.
Britain's parliament voted in favour on Saturday of an amendment that withheld support for Johnson's Brexit deal and as a result he is obliged, by law, to request a postponement.
The EU 27 have already agreed twice to postpone Brexit from the original deadline of March 29 this year.
FRUSTRATION IN BRUSSELS
However, frustration has mounted among them over the distraction of a process that has dragged on for three-and-a-half years since Britons voted in a referendum to leave the EU.
After the second extension they said it would be the last one, and French President Emmanuel Macron has been the most outspoken and hawkish among the 27's leaders on the issue.
His camp stresses the cost of protracted uncertainty in terms of sapping the EU's political capital and attention to face challenges from climate and migration to international crises, as well as economic cost for companies that have invested in contingency preparations for a no-deal Brexit.