The 21-year-old forward is being considered as an addition to the squad.
Chermiti moved to Everton from Sporting Lisbon in 2023 for an initial fee of €12.5 million.
He was brought to Goodison by Kevin Thelwell, who is now Rangers' sporting director.
According to The Rangers Review, preliminary talks have taken place with Thelwell's former club over the possibility of bringing Chermiti to Ibrox.
Rangers have been actively seeking new forwards after agreeing a £10.4 million deal with Lille for Hamza Igamane.
The Moroccan striker refused to play in the recent 1-1 draw with St Mirren and did not travel to Belgium for the Champions League exit.
The Rangers Review exclusively revealed on Saturday that Chermiti was under consideration should the club look to make more than one addition at No.9.
A deal with Girona has already been agreed for Bojan Miovski, with the 26-year-old striker set to join Russell Martin's squad.
The Rangers Review understands that the deal to sign Miovski is in the region of £2.6 million, with the potential for the fee to rise to above £4 million with add-ons.
The former Aberdeen striker moved to the La Liga outfit last summer following two impressive seasons in Scotland.
Rangers have also set a £5 million price tag for Cyriel Dessers, who is entering the final year of his contract next summer.
The club rejected an offer for Dessers from AEK Athens earlier this summer, but there is an expectation that their resolve could be tested for the 30-year-old in the window's final days.