Brendan Rodgers' side finished 12 points behind fourth-placed Arsenal last season, but have made a perfect start to the new campaign by beating Stoke at home before winning 1-0 at Villa Park on Saturday Night Football.
Former Reds midfielder Redknapp said they weren't at their best against Villa, especially in the second half, but says if they can keep hold of Real Madrid and Arsenal target Luis Suarez then they will become a formidable force when his suspension ends.
"Their ambition should be to get into the top four this year," said the Sky Sports pundit.
"The two Manchester clubs and Chelsea are a different class and I've said that before, but I think it's between Arsenal, Tottenham and Liverpool for that other place.
"They've got to bridge that gap. Twelve points is too much for a club of Liverpool's stature and they need to get a couple more players in.
"You can add the quality of Suarez if they manage to keep him. Iago Aspas looked ok today and I think he's a decent player, but when you get Suarez in they'll be an even better team.
"When you go away from home it's all about grinding results out. They weren't at their vintage best, but in the first-half they passed it really well and looked calm and composed.
"In the second half they did sit back a little bit and did try to hold on to what they had."
The game was settled by a moment of individual brilliance by Liverpool forward Daniel Sturridge, who beat several defenders and goalkeeper Brad Guzan to score with Liverpool's only shot on target.
And Redknapp sees him as a partner for Suarez, rather than a competitor for a place in the first XI.
He added: "Sturridge was an absolute nuisance for Okore and Vlaar and was always pulling into good areas.
"The relationship between Coutinho and Sturridge just gets better and better. His goal was a beautiful piece of individual brilliance.
"Brendan Rodgers will be well aware that when you bring Suarez back in, how do you fit them both into the system? But I always feel if you've got good players you can find a way.
"I think you could play Suarez in behind Sturridge in the role that Aspas played today. I don't see that as a problem."