Livermore, 21, bagged his first strike for the club on Thursday when he stroked home from 18 yards to score the third goal of the club's 5-0 Europa League play-off demolition of Hearts.
The Enfield-born midfielder looked calm and assured as he put on a strong display in Luka Modric's absence to maintain his impressive pre-season form.
Livermore has been sent on loan to no fewer than six Football League teams since he started his Spurs career in 2006, but he is now determined to stake a claim for a regular place in Redknapp's first XI, starting with Monday's trip to Manchester United.
Redknapp has already pinpointed Livermore as one of the players he intends to play regularly in the Europa League this season but a number of injuries in midfield may mean that he gets a run in the early stages of the club's Premier League campaign too.
The youngster admits that decision lies with Redknapp, but he is confident that he can deliver if called upon.
"Hopefully I can keep myself in the team now. I believe I am ready for the first team but it's just a matter of getting the chance," Livermore said.
"It's difficult to get in because we have such a strong squad throughout. It's hard for the manager.
"All you can do when you see your name on the team sheet is go out there and give it your all. You never know. We have a few injuries. I'll just have to wait and see."
Livermore broke forward regularly and linked up well with the rest of Tottenham's forwards in what was a brutal display of attacking football on Thursday.
Redknapp spoke in glowing terms about the midfielder after the match and another person impressed by Livermore was opening goalscorer Rafael van der Vaart.
"Jake did fantastically and that's always good to see," the Dutchman said.
"Luka is not playing at the moment but we had Livermore and (Niko) Kranjcar in there, he was fantastic too so it's a big compliment to those players."
Livermore, who spent last season on loan at Championship sides Leeds and Ipswich, is confident that he can add to his goal tally in the coming weeks.
"I scored a few times while I was on loan but it wasn't as many as I would have liked to," Livermore said.
"The way we play at Tottenham, where we keep the ball and break the opposition down, I find it easier to get forward.
"Sometimes in the Championship it can be hard to get in the game.
"I think I fit the way that Tottenham play and hopefully I can do well."
Livermore could keep his place in the team for the game at Old Trafford on Monday should Modric and Tom Huddlestone fail to overcome minor injuries.
Despite the champions looking in rampant form in the early stages of the season, Livermore believes Spurs can take at least a point from Old Trafford.
"It's a great occasion for us to go there and play against great teams like that so hopefully we can get a result," he added.
"We have a few injured players but they look like they are coming back and as a squad we are strong so I'd back us against anyone.
"Maybe we can go there and catch them cold."
Emmanuel Adebayor could be set for his Tottenham debut if he completes his loan move from Manchester City in time.
Livermore would welcome the Togo striker with open arms despite the fact that he used to play for bitter rivals Arsenal.
He added: "Adebayor is a great player. He would be a great addition to the squad. Robbie (Keane) has gone now so we need to strengthen up front a bit."
Source: Team Talk
Source: Team Talk