Intermediaries for the superstar duo have approached new Blues co-owner Todd Boehly to try to coax him into a sensational move for their clients.
It comes after Ronaldo told Manchester United that he wants to leave if they receive a suitable offer for him.
The 37-year-old is frustrated by a lack of transfer activity at Old Trafford and wants to play in the Champions League and fight for major trophies.
Ronaldo was allowed to skip the first day of pre-season training at United today due to family reasons.
His agent, Jorge Mendes, has already met Boehly and discussed a possible transfer for his client, as well as offering his services to move other players to the club.
Paris Saint-Germain are open to selling Neymar, and the Brazil international is understood to be open to a move to English football. But it remains unclear how Chelsea would pay his £606,000-a-week wages, with Thomas Tuchel reluctant to sign a superstar.
There are fears that a signing like Ronaldo or Neymar might disrupt the balance of the squad and would not suit their brand of high-intensity football. Tuchel is also keen to avoid a repeat of the mistakes they made signing Romelu Lukaku for £97.5million signing last summer.
Chelsea are growing in confidence that they will sign Raheem Sterling and hope to seal a £45m deal for the Manchester City star before they fly to the United States for their pre-season tour on Saturday.
But the Blues could be in the market for another attacker, with Raphinha stalling on their £55m bid to sign him from Leeds. The 25-year-old prefers a move to Barcelona and has told the Spanish giants he will wait for a move there, having already snubbed Arsenal and Tottenham
New Chelsea owners Boehly and Clearlake Capital have promised Tuchel at least six new signings this summer to ensure they are in a position to bridge the gap to Manchester City and Liverpool next season.