The 22-year-old is one of several players to have been told that they are not in Enzo Maresca's plans at Chelsea, and a move to Ipswich Town was initially thought to be on the cards.
Broja's proposed switch to Portman Road initially collapsed due to a failed medical, but Ipswich unexpectedly revived talks over a deal for the 22-year-old on transfer deadline day.
However, Ipswich and Chelsea ultimately could not reach an agreement over the striker's transfer, and according to The Athletic, Everton have now made an eleventh-hour push to sign him.
The report claims that the Toffees are hopeful of striking a deal that would see Broja join the club on loan for the 2024-25 season, with an option to buy for a sizeable fee of £30m.
Having got their deal sheet into the Premier League by the 11pm cut-off point, Sean Dyche's side have until 1am to complete the necessary paperwork, which they are hopeful of doing.
Everton are pursuing a move for Broja in spite of the fact that the former Southampton and Fulham loanee is nursing a foot injury, which he is expected to be recovering from until mid-October.
Broja's switch to Ipswich reportedly fell through for a second time due to Kieran McKenna's men not being prepared to accept Chelsea's demands for the injured forward, whose contract runs for another four seasons.
Broja joined the Chelsea youth system in 2009, but he has made just 38 senior appearances for the West London giants across all competitions, scoring three goals and setting up two more.
The Albania international follows Romelu Lukaku (Napoli) out of the Stamford Bridge door, but Chelsea have failed to add a new striker to their ranks this summer, as Victor Osimhen decided to stay at Napoli.