Di Resta was forced to bring his car to a halt after the tread came away from the left rear tyre during the early stages of a high fuel run in FP2.
Although Pirelli has not yet completed a full investigation in to what happened, initial analysis suggests that the tyre problem was the result of overheating of the tread rather than debris.
Of more concern to the Italian tyre company, however, is the way the tread appeared to come away, exposing the casing.
When asked by AUTOSPORT about the situation, 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.
"Visually it looks rubbish. We don't like that and we are looking at seeing if there are things we can do to eliminate that."
NEW CONSTRUCTION A FACTOR
Pirelli revised the construction of the tyres for this season, making the sidewalls stiffer as part of a bid to improve their performance.
Hembery believes those changes may have contributed to the strange failures, which includes Lewis Hamilton's problem in Bahrain practice.
"Absolutely," he said. "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."
Although di Resta's failure comes so soon after Hamilton's issue and Felipe Massa's double punctures in the same event, he thinks the situation appears worse than it is because of the way the tyres are failing.
"The mode of failure this year is much more spectacular visually," he said. "The tyre was still inflated from what I could see, so the structure did not give way. It was the tread. That is what we have seen in a few other cases.
"The tread has overheated and is coming away. From some point of view it would be better if the structure gave way and collapsed, as it would not look like quite so spectacular. In any case, it is not something we like and we are looking at what we can do."
Di Resta said that he was not unduly concerned about the tyre failure - which he said had been unexpected.
"It is not something we have experienced before, and we as a team are quite light on tyres," he said. "I don't' think it is anything we do with the tyres. Whether it is something to do with the compound I don't know."
Other drivers have raised more concerns though about the spate of failures though.
Sergio Perez explained: "With the tyre how it is at the moment, you do a couple of laps and then it is very difficult to have any feeling. It is a big concern for all of us.
"With two, three cars per weekend [hitting trouble], it can really be a serious accident."