The Daily Mail's Jack Gaughan described how De Bruyne and Sterling "will be offered an increase in salary to underline their commitment to the Premier League champions."
Both players are still contracted for another three years, but City are keen to secure them long-term while the club continues its appeal to the Court of Arbitration for Sport against a UEFA-imposed two-year ban from the Champions League.
City have been suspended from participation in the continent's most prestigious club tournament because of financial fair play irregularities.
Keeping De Bruyne and Sterling beyond the length of their current deals would represent a coup for City, particularly amid the ongoing battle with European football's governing body.
The Champions League trophy has proved elusive for City, and missing out on the competition for two years would damage the club's credibility as a burgeoning superpower in the modern game.
Domestic dominance has so far been the cornerstone of City's growing status, with De Bruyne and Sterling lifting two Premier League titles on Pep Guardiola's watch. De Bruyne's creativity has underpinned a free-flowing team, while Sterling's pace, timing and improvement as a finisher have often provided the cutting edge.
City won't make it a hat-trick of titles, despite having a game in hand on Liverpool, with the leaders still 22 points clear. Even so, Guardiola's side has already captured silverware by winning a third-straight Carabao Cup thanks to a 2-1 win over Aston Villa at Wembley.
Success in Europe remains the main goal, though, with City's focus naturally sharpened by UEFA's edict.
De Bruyne's vision and technique helped the Citizens build a 2-1 aggregate lead over Real. The Belgian crossed for Gabriel Jesus to head in City's opener, before tucking away a penalty himself to cap a classy performance in the Spanish capital.
Sterling came off the bench at the Santiago Bernabeu, and the England winger has reportedly emerged as a target for Real. He even told Mario Cortegana of AS how Real "are a fantastic club."
He added: "When you see the white shirt you know exactly what the club stands for, it's massive. But at the same time, I have a contract with City now and I have to respect this. But it's a fantastic club."
Still just 25, Sterling represents not only City's trophy-laden present, but the promise of future glories. Meanwhile, 28-year-old De Bruyne remains the heartbeat of the team and is a player just approaching the peak of his powers.
Retaining both would keep City among Europe's elite, with or without a place in the Champions League.