Tahir, the second-leading wicket taker at the tournament, suffered the injury while taking a return catch to dismiss Jonathan Trott en route to figures of four for 38 in Chennai.
The 31-year-old Pakistan-born leg-spinner has been a revelation at the tournament having taken 11 wickets at 8.90 after making his Proteas debut in their opener against West Indies.
His participation in the remainder of the global showpiece is, however, now in some doubt with Cricket South Africa revealing today they would make a decision on his fitness in the next two days.
"Imran injured himself during the game against England while attempting the caught and bowled, which dismissed Jonathan Trott," Proteas' manager Dr Mohammed Moosajee said in a statement.
"The x-ray that was done today revealed a small undisplaced fracture in his left thumb.
"The images have been sent home to Cricket South Africa's medical committee and we are waiting for the hand specialist, Dr Mike Solomons, to give us an opinion.
"His thumb has been splintered and immobilised and a decision on what happens further will be made in the next two days."