According to the Daily Star's David Woods, the pair will be hoping to take advantage of the Blues bringing in further midfield reinforcements this summer, which could see Fabregas' struggles for game time intensify.
The Spaniard is aware of interest from United boss Jose Mourinho, who signed him for Chelsea in 2014, and he may be encouraged by Liverpool manager Jurgen Klopp's ambition to recruit him, too.
James Pearce of the Liverpool Echo offered a contrasting report, though, as "Anfield officials have rubbished" the speculation linking him with the Reds.
However, Liverpool and United could offer him the chance to do that and enjoy more first-team football if they do pursue him.
Fabregas, who will turn 30 on Thursday, has played just 1,642 minutes in all competitions this season but has nevertheless managed to rack up six goals and 12 assists.
As Squawka Football demonstrated, the midfielder is among the Premier League's most creative talents:
Fabregas specialises in breaking through defences, per football journalist Dan Levene:
Creativity isn't a quality either Liverpool or United lack. According to Squawka, the former are second only to Tottenham Hotspur in chances created this season with 447, while the latter are fourth with 419.
While United have been slightly less creative, they also perhaps have less need for Fabregas.
With Paul Pogba and Ander Herrera already in the side, the Red Devils have a more pressing need for a holding midfielder. Fabregas could operate as a deep-lying playmaker, but there is a question mark over how much protection he would offer the back four.
At Anfield, Fabregas would be an upgrade on their midfield options, and Liverpool's front three of Sadio Mane, Roberto Firmino and Philippe Coutinho would be able to thrive with his vision and passing range behind them.
The Spain international is a classy operator and would be a tremendous signing for the Reds. Given his ability, United might still pursue him if he's available this summer simply because it's not often a player of his quality is attainable, though much also depends on Chelsea's willingness to sell to one of their rivals.