The Irish striker scored 122 goals in 306 appearances across two spells at White Hart Lane.
Tottenham currently sit in fourth spot in the table after 22 games ahead of Saturday’s trip to Crystal Palace, five points behind league leaders Arsenal and two points above fifth-placed Manchester United.
Keane, who enjoyed two spells at White Hart Lane before leaving for the LA Galaxy in August 2011, believes the club are making great strides under head coach Mauricio Pochettino but thinks that this season might have come too soon for a title challenge.
The 35-year-old Irishman has seen too much inconsistency from the Argentine’s young squad over the first half of the campaign, but he insists that a top four finish would still represent good progress.
Asked about Pochettino’s management, Keane said: “I think he’s doing a very good job. They’re quietly starting to creep up on teams. People mention the likes of Manchester United and Liverpool, but Pochettino’s gone about his job in a nice, quiet way.
“But the problem with Spurs is they win two or three games on the bounce and then play Leicester at home and lose. If they want to push for the Premier League, I think it’s probably a bit too soon for them.
“If they got into the Champions League it’d be a big achievement, and I think they certainly have a good chance of that. They’ve got good players in the team like Harry Kane, whose doing a tremendous job.”
Nevertheless, Keane still thinks that Pochettino’s side need to make a couple of attacking signings this month to take some of the goalscoring burden off Kane’s shoulders.
Spurs only have Son Heung-min available as cover for the lone striker role currently, and the former Liverpool man worries that their lack of depth could come back to bite them in the event of injuries.
However, with Kane an ever-present in the league so far this season, he also acknowledges that it will not be easy to find an establish goalscorer in January who is happy to play second fiddle to the England international.
“I think it’s important that they get in a couple of players during the window to help Harry, but I certainly wouldn’t get someone in to take minutes away from him,” Keane told ESPN FC. “It’s very difficult to consistently keep Harry playing and have someone in case he gets injured to come in and score goals.
“But at the same time, you can’t rely on one striker for the whole season. The others they have don’t score enough goals. If Harry gets injured and is out for a few weeks, who do you go to? That’s probably Tottenham’s main problem.”