The BBC has learned both sides in the dispute came "tantalisingly close to a deal" late last Tuesday.
The Communication Workers Union (CWU) said up to 120,000 workers "solidly supported" this week's action, which Royal Mail called "unnecessary".
Royal Mail described the plans for three further walkouts on Thursday, Friday and Saturday as "appalling".
Managing director Mark Higson urged the CWU to accept an agreement on the table on Tuesday.
"Signing that agreement... will mean that next week's strikes will be unnecessary and, more than that, we will not have any more action this side of Christmas," Mr Higson said.
BBC business correspondent Joe Lynam said the negotiations in Windsor on Monday and Tuesday between a team from Royal Mail and trade unions resulted in an agreement on a form of words which would have led to the strikes being called off.
He added the CWU had said a letter on Wednesday morning from Mr Higson scuppered any putative deal between the sides.
However, the union has described the suggestion that wording had been agreed as "total nonsense".
It told the BBC, "significant progress was made on issues which would have called off the strikes," in particular on local issues, but "the executive committee had not been asked to make a decision on any form of words which would have ended the dispute".
About 78,000 delivery and collection workers walked out on Friday following Thursday's strike by mail centre staff.
A Royal Mail spokesman, when asked whether the backlog would clear before the next strikes, said: "Our aim will be do do that. We will have to see how things go."
Meanwhile the GMB union said it had started to receive calls to a hotline it has set up for members of the public to report employment agencies supplying staff to Royal Mail during strikes.
The unions have said the practice is illegal, but Royal Mail has denied its move to hire 30,000 agency workers to deal with the effects of the strike as well as the Christmas rush, broke the law.
If they go ahead, the strikes next week could involve:
• Thursday - 43,700 staff in mail centres, delivery units in mail centres, network logistic drivers and garage staff walking out from 0400 GMT
• Friday - 400 workers at three sites in Plymouth, Stockport and Stoke, who assist mail centres by reading and entering mail addresses
• Saturday - 77,000 delivery and collection staff across the UK.
Meanwhile, HM Revenue & Customs has said it will not extend the 31 October deadline for filing tax returns because of the postal strike, but added that any returns received late because of the strike would be unlikely to incur a fine.
The public give their views on the Royal Mail strike action
Last year 2.4 million people sent returns by post.
If people miss the postal deadline, they can still file online by 31 January.
Consumer groups such as Which? and Consumer Focus have also called on companies to be lenient with those who are late paying bills because of the postal strike.
Mike Wildman, from Liverpool, who runs an internet mail order company told the BBC his sales were down 90% this week.
"People are just not mail order shopping as they think they won't get their parcel," he said.
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