The 22-year-old is understood to have fallen ill over the weekend and been admitted to St Mary's hospital in Paddington for tests.
Sturridge's condition is said to have improved over the last 48 hours, but his place in the 23-man
Stuart Pearce's squad is set to meet at the Team GB holding camp in Loughborough next weekend, and Football Association medical staff will assess the results of tests before deciding if Sturridge can still play a part.
Pearce has until 25 July to select a replacement should Sturridge be forced to withdraw, with those on the stand-by list believed to include Huddersfield Town striker Jordan Rhodes.
But immediate concerns revolve around the health of Sturridge, with more details of his condition expected when Chelsea boss Roberto Di Matteo holds a routine pre-season press conference at the club's training ground on Wednesday morning.
Doctors reportedly suspect Sturridge may be suffering from the viral strain of the disease, whose symptoms are generally mild and which can be overcome without the need for medical treatment.
Sue Davie, chief executive of the Meningitis Trust, told Sky Sports News: "I can't comment on specific cases but if it's viral meningitis it's a less severe form, although it can be very debilitating.
"A sufferer can have symptoms such as headaches, tiredness, a stiff neck, nausea, vomiting and a dislike of bright lights. It can go on for months.
"It can't be treated with antibiotics very much. It's about resting and getting proper care in hospital, being looked after well and making a full recovery.
"But it will take a little while. I would imagine his doctors will say to him he needs to rest and take care."
Newcastle Jets have been crowned Champions of the Singapore Soccer Sixes after defeating Chonburi FC of Thailand 4-1 in Sunday’s grand final.