Although the governing body took Mercedes to the International Tribunal for its running of a current car at Pirelli's post-Spanish Grand Prix test, it is understood that it is now prepared to ease testing restrictions on safety grounds.
Pirelli already has two private tests lined up - one at Paul Ricard next week and the other at Barcelona a fortnight later – which it is currently scheduled to do with its 2010 Renault test car.
Following a meeting between FIA president Jean Todt, F1 supremo Bernie Ecclestone and Pirelli motorsport director Paul Hembery at Silverstone on Saturday, it has been agreed that testing restrictions need easing to help the sport's tyre supplier.
It was at that meeting – and not after the race – that Todt invited Hembery to take part in this week's Sporting Working Committee meeting to help formulate tyre testing plans.
AUTOSPORT has also learned that at that meeting, Todt informed Pirelli that he accepted the FIA needed to do more to help the tyre company's testing needs.
There were also assurances made that if there were sufficient safety grounds, then the FIA would allow Pirelli to do whatever it took to solve them – even if that meant calling upon a team to provide a 2013 car.
Speaking about Todt's offer before the British GP, Hembery told AUTOSPORT: "That is really good. It shows we have all learned from experiences and there is no point repeating bad experiences.
"It is important you make changes going forward and that is something that Jean suggested on Saturday, which is very positive."
Hembery said that relations with the FIA had improved dramatically since the Tribunal hearing, where Pirelli reserved the right to take the governing body to the civil courts.
"Sometimes you have to have these situations to make change," he said. "All of us have to look at what we did and make things better.
"If the real outcome of this is that we do get some positive changes then it has been worthwhile."
While Pirelli had previously invited teams to join the tests, none had expressed any interest in doing so.
In the wake of the Mercedes trial, there remain issues about the legality of testing with a 2013 car. AUTOSPORT revealed last weekend that Red Bull had written to the FIA seeking clarification about what is and is not allowed.