Recently, the FIA has been tendering is contracts with F1's sole tyre suppliers on a three-year basis, and Pirelli is now approaching the end of its first run.
Hembery said Pirelli wants to stay on the grid beyond 2013, but is pushing for an even longer contract than three years.
"We will know soon if we get a new contract as the sole supplier in Formula One," he is quoted by the German-language Speed Week.
"We would prefer a long term contract, say five years," he added, referring to the world of Nascar racing, where Goodyear's latest contract is for five years.
Asked when Pirelli needs an answer from F1 about the future, Hembery smiled: "Yesterday, ideally.
"Seriously, we have 600 professionals at work, and so if we're no longer in F1, then we have to see how we can use them elsewhere.
"Unlike last time, we don't want the decision in June or July -- it's just too late," said the Briton.
Complicating the matter further, added Hembery, is the switch to turbo V6 power for 2014.
"It is expected that the turbos will develop more torque, which influences the traction of the corners," he said.
"More traction means more temperature, and we also need to know what to expect in terms of top speed so that we don't have problems with the integrity."
After a sneak peak of Pirelli's 2013 tyres in Brazil late last season, the F1 field got their first real taste of this year's rubber at Jerez on Tuesday, as the official pre-season test period began.
Red Bull's Mark Webber - and many other drivers - complained about high wear, according to Germany's Auto Motor und Sport.
"You could see it with the naked eye, so sudden was the loss of grip," said the Australian.
"The difference between a new and an old tyre was very big. After a few laps, the times were falling off dramatically."
Webber added: "But Jerez is a bit different. You can have problems here that you don't have somewhere else."
Romain Grosjean agreed: "The asphalt here is very rough and the layout is hard on the tyres, and the temperatures are also low."