Redknapp is the overwhelming favourite to succeed Fabio Capello as England manager after the Italian's shock resignation on Wednesday.
The Spurs boss admits he considers the position as the "ultimate" job in football for any English manager and he did his chances no harm by masterminding a resounding 5-0 hammering of Newcastle on Saturday.
A double from Louis Saha on his full debut, and strikes from Benoit Assou-Ekotto, Niko Kranjcar and Emmanuel Adebayor helped the Londoners stroll to victory with a breathtaking performance full of flair, power and pace.
Tottenham's supporters made their feelings towards Redknapp clear by chanting "We want you to stay" on several occasions, and King echoed their sentiments on Sunday night by saying: "As a group of players we love him and hopefully he will stay.
"This sort of speculation always happens in football, but we would love to keep him.
"Harry has brought the club on massively. We were bottom of the league when he first came in and the results picked up immediately.
"Straight away he showed his people skills, his ability to lift players even when their confidence was low.
"He has gone from strength to strength, brought in some good players, experienced players who know what it takes. It has been really good."
The Football Association met on Friday to draw up a shortlist of possible candidates to replace Capello and after being cleared of tax evasion on Wednesday morning, there now appears to be little preventing Redknapp from taking the job.
Redknapp is likely to remain in his current role until the end of the season, however, after admitting on Friday that it would be almost impossible to combine the two roles.
Having seen the club climb to third in the Barclays Premier League table, King is determined not to let talk of Redknapp taking the England job distract him and his team-mates in their quest for a first title since 1961.
"We need to keep our focus until the end of the season. What will happen in the summer will happen," the defender said.
"I am pretty sure the gaffer will be here until then."
Assou-Ekotto put Tottenham ahead with a good finish from a tight angle on Saturday before Saha added his first from an Adebayor cross after five minutes.
The Frenchman then added a second after 20 minutes before goals from Kranjcar and Adebayor sealed the rout for the hosts, who are now 10 points clear of fourth-placed Arsenal.
Saha admitted afterwards that his first start had a Roy of the Rovers touch about it.
"It was a great, great feeling," the 33-year-old said.
"I am not that young, but it's just like I was living the dream of a young boy - to make your debut and score on your home ground, it's just amazing.
"As a kid, you always dreamed of playing fantasy football, attacking football and you've seen that here.
"In the first 15 minutes, it was like the quickest football that I've ever played. To be part of that is just a pleasure."
Newcastle made the long trip back to Tyneside having barely threatened Tottenham goalkeeper Brad Friedel.
Demba Ba created a couple of chances, but in all honestly Newcastle looked way out of their depth despite sitting just three places below Spurs in the league.
"Spurs caused us all sorts of problems," manager Alan Pardew said.
"We conceded two quick goals and we couldn't get ourselves settled after that.
"We had a really bad opening 15 minutes where we were all over the place.
"Every one of their players played absolutely brilliantly. I can't think of one who had an off day.
"I'm pleased we stayed disciplined in the second half, though.
"We didn't really let them expose us too much even though they had their tails up and were playing with great confidence.
"That is about the only thing we can take out of it really."