Vardy's two goals in Saturday's 4-1 win at relegated Huddersfield lifted him level with Lineker's total of 103, joint sixth on Leicester's all-time list.
The current Foxes forward thought he had surpassed Lineker's mark, but Leicester have not counted Vardy's goal against Manchester United in the 2016 Community Shield.
"I've just been told I overtook him?" Vardy said. "But hopefully there's more to come and we know that's only going to happen by putting more hard work on the training field first.
"There's still work to be done. It's early days. We're showing signs of improving and we want to keep going and obviously make sure performances are consistent."
Vardy has hit top form under Brendan Rodgers, who replaced former boss Claude Puel at the end of February.
The England striker has notched seven goals in his last six matches, while Leicester's fourth straight win on Saturday lifted them up to seventh.
"The manager's got us playing now the way he wants us to play. It's benefiting each and every one of us and I think you can see that.
"We'll keep working hard at it and keep putting the time and effort in on the training field to take it into the game.
"It's been brilliant on the training field. He's getting his ideas across and everyone's buying into it and putting the effort in to make sure we're playing the exact way he wants."
Vardy applied a clinical finish early in the second half after Youri Tielemans had fired Leicester into an interval lead.
Aaron Mooy's 52nd-minute penalty gave Huddersfield fleeting hope, but James Maddison's free-kick and Vardy's spot-kick in the closing stages sealed a comfortable win.
Huddersfield, relegated after last week's defeat at Crystal Palace, remain bottom and slipped to a ninth loss in 10 matches under boss Jan Siewert.
Siewert is expected to overhaul his squad in readiness for the Sky Bet Championship, but midfielder Jon Gorenc Stankovic is determined to stay and help them bounce back.
"I still have a year left on my contract and we will try to make a challenge to return to the Premier League," the 23-year-old Slovenian said.
"The Championship is hard and everyone can beat everyone, but we will give our best and try to come back again."
Mooy's penalty was just Huddersfield's eighth goal in 17 home league games this season and Stankovic could not hide his disappointment after their latest defeat.
"It has been difficult to keep spirits up knowing we are relegated, but we try to stay in a positive mindset," he added.
"I know we are in danger of setting the lowest goals record, but we have been scoring recently and I believe there will be more to come, so that is a positive."
Join our new commenting forum
Join thought-provoking conversations, follow other Independent readers and see their replies