The Liverpool star suffered the injury in training on Friday and has been taken to hospital to have a scan.
The news is a concern for boss Roy Hodgson just three days before the qualifier.
A post via the team's official Twitter account, @england, said: "Daniel Sturridge felt an injury in training this morning and will go for a scan this afternoon. He will then be assessed by England medics."
Gary Cahill sat out Friday's training session but the prognosis on him is more upbeat - he is expected to recover in time for the Swiss showdown.
The Chelsea player, 28, limped off late on in England's tepid 1-0 win over Norway at Wembley on Wednesday night with an ankle problem - but manager Roy Hodgson is confident the injury will not keep him out of the Group E opener in Basel.
Although Cahill was unable to take part in Friday's session at St George's Park, he is expected to return to training on Saturday and take his place at the heart of England's defence when they face what, on paper at least, looks their sternest test of the campaign.
Should Cahill fail to regain fitness in time, it is likely that Hodgson would opt to call on the experience of Everton captain Phil Jagielka paired with Manchester United utility man Phil Jones.
With West Brom goalkeeper Ben Foster and uncapped Newcastle midfielder Jack Colback already pulling out through injury, only 19 of Hodgson's 20-man squad took part in Friday's workout.
As a result, several of England's U20 squad made up the numbers in the session, with Southampton midfielder Harrison Reed, Barnsley's Paul Digby and Exeter goalkeeper Christy Pym all joining in ahead of their clash with Romania in Telford later in the day.
Watch highlights of England's opening European Qualifier away to Switzerland on Sky Sports 5 HD at 10pm on Monday night.
At least Welbeck can fill the profligacy role. Otherwise I'm not sure how we'd cope!- de gea bar