According to the Mirror's Darren Lewis, the Blues have joined Barcelona in being interested in the 21-year-old, who has a £44 million release clause.
The Catalan club are further along in their pursuit of Arthur, but they may have put the deal in jeopardy after the player was photographed in Barcelona colours alongside club officials, per AS:
Andre Cury, who helped the Catalan club bring Neymar to the Camp Nou from Santos, is also said to have had dinner with some of the player's representatives, prompting Gremio to describe Barcelona's approach as "unprecedented."
Arthur confirmed the talks with the Blaugrana: "On the Barcelona matter, they did speak with me. We had a meeting, but I want to say that there's nothing final. Nothing has been defined, and I did not sign anything. I am a Gremio player."
As the Brazilian club have now refused to negotiate with Barca, it seems they will have to meet his release clause if they are to land him, but Chelsea may well hope to secure him for less.
Fabregas' place at Stamford Bridge appeared to be in jeopardy this time last year as manager Antonio Conte largely restricted him to the bench, but the Spaniard has turned things around and has already made 20 starts this season having made just 18 last season.
Conte opened up on his situation and discussed the midfielder turning 30, per ESPN FC's Liam Twomey:
Liam Twomey @liam_twomey
Conte asked how long Fabregas can go on for: "When you’re 30 you have at least five years to play football. For sure to play at this level you must show great desire, because it’s not simple. You have a lot of competition but Cesc has great ambition. He’s a winner in his mind"
Fabregas is hardly at the stage where he needs replacing, but it's a good move from Chelsea to be considering long-term successors for when he does start to decline.
Though he has no experience of European football, Arthur could be a strong choice given his talents lie in controlling matches and dictating the tempo in the middle of the park, which is what Fabregas can bring.
Gremio recognise his future lies elsewhere but hope to keep him for the immediate future. Per Lewis, club vice-president Odorico Roman said:
"I'd say it's rare to find midfielders of Arthur's level in the world right now.
"There is a €50 million release clause, but if a club really wanted to sign him, they'd have to send us a proposal. We would evaluate and decide whether to accept or not.
"Our intention is to keep Arthur with us for a little while longer. We think he can remain for another one or two seasons and give a big contribution to our cause."
Staying would allow him to continue playing regular football and develop his game further—making him even more suited to taking over from Fabregas a year or two down the line—but waiting that long would hand the initiative to Barcelona, who could move to snap him up in the meantime.