Pirelli is currently planning to run an experimental 'ultrasoft' tyre in Friday practice in Singapore and Abu Dhabi as it prepares for 2016.
From next season, the FIA is expected to allow Pirelli to have extra compounds in its range, and the ultrasoft – also known by the company as a "super supersoft" – will provide extra flexibility.
Under current rules teams can run experimental tyres on Fridays.
"Hungary is probably too early," Paul Hembery told Motorsport.com. "It will probably be in Singapore and Abu Dhabi as the Friday experimental tyre.
"But these things change all the time, there's been another proposal and we may be running something a little bit more bespoke."
One F1 team insider suggested that the ultrasoft was essentially last year's supersoft, but Hembery insists it isn't: "That's not true. We've got five or six potential solutions."
Meanwhile, Pirelli is in discussions with the teams about a test in Abu Dhabi in the days after the final race.
"Obviously the teams have got limitations on their running times, they've got engine limitations as well, and you don't want them to be risking anything during a competitive part of the season.
"Abu Dhabi could be a solution, running afterwards. It's the most suitable time probably, because nobody is fighting for the championship. We'll do what's needed to do the job."
However, there is no provision in the rules for such a test, and there is a debate about the cost if it is purely a tyre test, with the teams expecting Pirelli to foot the bill.
They have told the Italian company that the test would cost them £600-700,000.
"Then we just need one car to do, we don't need 10 cars!," said Hembery.
"They [the teams that don't test] are going to have a pre-season test anyway, it's only a screening process, and you're not really going to have much of a benefit. They will still have plenty of running."