Redknapp was reported to have held talks with chairman Daniel Levy on Wednesday amid claims the 65-year-old would leave White Hart Lane with a year to run on his contract.
Redknapp called speculation he was on his way out as "scandalous" on Tuesday night, but despite that reports have now intensified with some claims he could leave by the end of Thursday.
Those reports have been further fuelled by a number of bookies closing betting on Everton boss Moyes being named as his successor.
Negotiations over a new deal for Redknapp have stalled since the start of the year when Redknapp, who guided Spurs to fourth place in the Barclays Premier League last season, was linked with the then vacant England manager's job.
In the past week Redknapp has made it known he would be uncomfortable heading into the new season without a longer deal while also defending his relationship with Levy.
It has been claimed, however, that Levy was only willing to offer Redknapp a rolling 12-month contract whereas the former Portsmouth manager is thought to want a longer deal that will make the London club his last job in management.
Redknapp did achieve Levy's goal of reaching the top four last term, but Spurs missed out on playing in the Champions League after Chelsea's penalty shootout win over Bayern Munich last month.
An exit may seem harsh on Redknapp but his critics point to the fact that his team would have qualified directly for Europe's premier club competition had they not thrown away a 10-point lead over Arsenal towards the end of the season after a poor run of form.
Redknapp vehemently denied he was on his way out on Tuesday night as he told ESPN: "No, I haven't resigned, and I have no idea why it is being suggested that I have resigned.
"This is an outrage; an absolute liberty for people to be putting around this kind of rumour on the internet.
"It is not true, there is not a chance I will resign. Why should I?
"I have a year left on my contract."
Earlier this week Redknapp denied suggestions he was being courted by a club in Qatar after telling Tottenham to speed up negotiations.
"The simple situation is, I've got a year left on my contract. It's up to Tottenham whether they want to extend that contract or not," he said.
"If they don't extend it and I go into my last year, it's not an easy one when players know you've only got a year left.
"If Daniel doesn't think I'm worth it, that's up to him, that's up to the club. There's nothing I can do about it."
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