The Scot was forced to bring his Force India car to a halt after the tread completely came off the left rear tyre during the second free practice session ahead of Sunday's Spanish Grand Prix.
An initial analysis suggested that the problem was caused by overheating of the tread rather than debris on the circuit.
Pirelli motorsport director Paul Hembery said: "It doesn't look pretty when you get the tread coming away like that. If anything like that happens, it tends to collapse the tyre on to the rim, so it did not look good. This year we are seeing the tread coming away.
Di Resta's frightening experience was not the first of its type this season.
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Lewis Hamilton suffered a similar incident during practice with Mercedes at the Bahrain Grand Prix.
Hembery said that he believed a new system of construction of the tyres this year -- with the sidewalls made much stiffer -- had contributed to this particular type of failing.
"That has pushed the failure into a different type of conclusion, so we are looking at what we can do under the constraints that we have," he explained. "We dont like it and we are taking a very close look at it.