The German winger is a wanted man this summer, with Premier League rivals Manchester United and Man City also linked with an interest.
According to the Daily Mail, the Blues are now also considering a move for the Bayern Munich man as they look to strengthen their attacking options.
However, reports in Germany suggest Chelsea have already suffered a big blow in the early days of their pursuit.
German outlet Bild claim the former Arsenal man wouldn't consider a move to Stamford Bridge due to his allegiance to the Gunners - and would even prefer a return to the Emirates if he were to leave Bayern this summer.
Gnabry spent his formative years with Arsenal and made 18 appearances for the north London club before leaving for Werder Bremen in 2016.
A successful year there earned him a move to Bayern Munich in 2017, where he has since scored 64 goals and provided 40 assists in 171 appearances.
But, despite his record for the club, the 26-year-old's future with the German champions is far from certain.
Gnabry has just one year left on his contract at the Allianz Arena and his long-term future is uncertain after he reportedly rejected a new deal worth around £200,000-a-week to extend his stay.
Man United and Man City are among the clubs interested and, while Chelsea are now also keen, it seems like a move to the west London club remains unlikely.
The Blues have already agreed a deal to sign Raheem Sterling from Man City for a reported fee of £55million.
Thomas Tuchel was also linked with an strong interest in Raphinha, however it seems that has since cooled.
Barcelona are said to be closing in on a move for the Leeds United man, however, talkSPORT understands they're hoping to agree a deal that would see the cash-strapped Spanish giants put off paying the first instalment until next year.
While nothing is confirmed, it seems that Raphinha's move to Chelsea is hanging by a thread and, with Gnabry reluctant to move to west London, it means Chelsea's search for reinforcements continues.