The 22-year-old went down in the first half of the match at Ibrox after colliding with pitchside boards, and left the stadium in a boot and knee brace.
His injury came hours after it was revealed Rangers had rejected a big-money bid for the player from Galatasaray.
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Sky Sports News understands the club are also looking to add another forward during the final weeks of this transfer window, following Chermiti's injury.
While several options are being looked at, it is understood no talks are ongoing over a move for Ryan One of Sheffield United, despite reports.
Chermiti, capped at U21 level by Portugal, joined from Everton last summer in a deal worth up to £10m and scored 15 goals and provided five assists across all competitions last season.
That has led Ibrox chiefs to value him higher than the club-record fee they received for Calvin Bassey, who joined Ajax for an initial £19.6m in 2023.
Speaking after Sunday's defeat to Hibernian, which left Derek McInnes winless in his first three games as Rangers boss, he said: "Disappointed for the big fella that he has to come off on a stretcher. I thought he was looking bright and running gaps.
"He played with a bit of purpose and there's no doubt we missed that, but he's left in a boot and his knee in a brace.
"It's no surprise that he's a player who's got a big tag on him and he's got high potential, but thankfully he's our player and that will continue to be the case."
Rangers in talks for Meghoma as Driouech closes in on £6.5m move
Rangers have made nine signings this summer, and that number is set to rise this week.
The club are in talks with Brentford over a deal to sign Jayden Meghoma on loan.
Sky Sports News understands that if terms can be agreed, he could be in McInnes' squad to face St Mirren in the League Cup on Sunday.
Meghoma, 20, spent last season on loan at Ibrox and made 44 appearances.
Meanwhile, Rangers are assessing Tuur Rommens' fitness ahead of the second leg of their Europa League third round qualifier against Jagiellonia Białystok.
As well as Meghoma, the club are still looking for other reinforcements in defence.
Talks are ongoing with Aston Villa over right-back Kosta Nedeljkovic. Meanwhile, it is thought Hugo Siquet of Club Brugge is keen on a move to Ibrox - with Rangers in talks over a possible deal for the 24-year-old.
Chairman Andrew Cavenagh confirmed the club will add in the region of three to five more players before the transfer window shuts on September 3 in Scotland.
Winger Couhaib Driouech will be the first of those, and become the club's 10th arrival this summer.
The 24-year-old is completing his medical and paperwork ahead of a £6.5m move from PSV.
Rangers have beaten off strong competition from various clubs, including Burnley and Celta Vigo, for the player.
Driouech scored 12 goals in 61 games for PSV, including three in the Champions League last season, plus provided 10 assists.
Meanwhile, Mohamed Diomande is expected to be the next outgoing, with Turkish club Corum FK set to sign the midfielder for around £4.5m.
