Topley rolled his left ankle during some catching drills in Brisbane on Monday afternoon and did not feature in their final warm-up against Pakistan.
England announced Topley "will be assessed throughout this week" and are hopeful of a swift recovery.
It is an unwanted headache as they travel to Perth to take on Afghanistan on Saturday, with Topley becoming an increasingly dependable figure in the white-ball ranks since the turn of the year.
The left-arm seamer has taken 17 wickets in 16 T20s in 2022 and can bowl anywhere in an innings, with an economy rate of 7.8 impressive given he operates mainly in the powerplay at the death.
Curran and David Willey are England's other left-armers in their 15-strong squad so they are unlikely to this weekend risk Topley, who has overcome multiple stress fractures in his back during his career.
Curran is expected to be in England's XI against Afghanistan, irrespective of whether Topley features, and believes they are tracking well after an improvement in results recently following a disappointing summer.
England recorded a 4-3 series win in Pakistan and then thrashed the same opponents by six wickets in a low-key practice match on Monday, having also beaten World Cup hosts and defending champions Australia 2-0.
"We're definitely ready now," Curran said.
"We're feeling like we're almost not peaking but we're playing good cricket, (as shown in winning series against) Pakistan and Australia then this game.
"We're really excited, hopefully with a couple of good training sessions in Perth, come the weekend we'll be good."
Curran, who was ruled out of last year's World Cup with a stress fracture in his lower back, demonstrated his all-round ability with an unbeaten 33 off 14 balls against Pakistan.
Moved up to No.6, Curran showed he is quite adept at the finishing role with a cluster of boundaries.
"It's very challenging but it's just a role you've got to enjoy," Curran said.
"Guys like Moeen (Ali), (Liam) Livingstone, Harry Brook are experienced in that middle-over phase so they help me through it.
"There will be some days where it doesn't work, some days where it does."
While England go into the World Cup in buoyant mood, Curran warned against complacency given the first two days have already produced two shocks, with Scotland upsetting West Indies, the 2012 and 2016 champions, a day after Namibia stunned Asia Cup winners Sri Lanka.
"It's a T20 format, anything could happen," Curran added.
"Afghanistan are a side that are extremely skilful ... We know it's going to be a really tough opening game."
7°C