Ahead of the festive matches against Liverpool and Manchester City, all the Italian would say was that if his team won both they would have 44 points, four more than the first objective he had set, to avoid relegation.
Leicester, who avoided going down in May thanks to a late run of victories under Nigel Pearson, began the season at odds of 1,000-1 to be top of the Premier League at Christmas.
That encouraged one optimistic fan to wager five pounds on them with bookmakers Ladbrokes.
The supporter celebrated after a 3-2 win at Everton on Saturday helped Leicester go into the holiday fixtures with a two-point lead over second-placed Arsenal.
Ranieri, who took over in the close-season after Pearson was sacked, said he was looking no further than Saturday's return to Merseyside to play Everton's neighbours Liverpool.
"We are doing well but haven't achieved anything yet," he told reporters. "If we win the next two games we have 44 points, I hope with 40 we are safe.
"I'm proud of my players and our fans. The fans are dreaming and I don't want to wake them up."
Ranieri, back in English football after being sacked by Chelsea in 2004, has been impressed by the spirit at the club, exemplified by seeing injured midfielder Danny Drinkwater sitting among the visiting fans at Everton.
"I don't know if it was our best (performance)," he said of that win. "I hope our best is the next."
Ranieri also addressed the question of Leicester's direct style after statistics released this week showed they play fewest sideways and backwards passes of any Premier League team, and also take fewest passes before every attempt on goal.
"Possession is important but it's not our best [quality]," he said. "We're strong because we play to our strengths."
Those strengths have made Leicester the team of the season so far with two of the league's outstanding performers in top scorer Jamie Vardy (15 goals) and Algerian Riyad Mahrez (13).
(Reporting by Steve Tongue; Editing by Tony Jimenez)
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