The Nigerian and his back-line team-mates are hoping to help Hoops' keeper Fraser Forster wipe out Aberdeen goalkeeping legend Bobby Clark's 43-year-old record of 1,155 minutes without letting in a top-flight goal.
The Celtic shotstopper equalled the Scottish clean-sheet record with his 12th consecutive unblemished performance in the 3-0 SPFL Premiership win over St Johnstone last Sunday.
Now Forster is just 31 minutes short of breaking Clark's record while, should he keep the shutters up at Tynecastle for more than 87 minutes, he will break the UK record currently held by Dutch goalkeeping master Edwin van der Sar from his time at Manchester United.
But although Ambrose admits everyone is talking about the potential of Forster, he claims Celtic eyes will remain firmly on the business of taking another vital step towards the first-ever SPFL Premiership title.
He said: "Everybody is talking about it but it is not easy to achieve.
"It has taken a lot of hard work, dedication and commitment. It is close but you never know when it is going to end. We just think 'game after game' and see what happens.
"I'm sure the fans will let us know if we do it but we just have to keep it going. But I think it will be tough against Hearts this time and we must stay focused on winning the game.
"I definitely think it will be different to the previous game at Tynecastle (Celtic won 7-0).
"After what happened the last time, they will give us a good fight. They won't want to repeat it and we will want to win, so it will be a real fight.
"But we are Celtic and we know how to win games and first of all that is what we must try to do tomorrow."
Despite the fact that Forster is on the verge of creating history along with his defensive cohorts, the England international's hopes of going to the World Cup this summer hang in the balance.
But Ambrose has no doubt that the Celtic No.1 should be a cert for Brazil and dismissed suggestions that playing in Scotland is damaging his team-mate's World Cup chances.
He said: "I believe Fraser is different class. Nobody can say there is anything he can't do because already with Celtic he has done everything.
"He has kept clean sheets and, in every game he's played, he's been in top shape.
"Week in, week out he's been doing it and I believe he should be given a chance to play in the World Cup because I don't think there is much difference between playing in Scotland and the Premier League in England.
"We have played against the best in the Champions League and Fraser has kept Celtic going.
"He should go to the World Cup and it doesn't matter if he's playing in England, Scotland or Germany. What he is doing is special.
"I believe in a couple of years he will be the best in England. He can be the best in the world and right now he is one of the best."