"Time is of the essence, I would suggest," the Italian company's motorsport director Paul Hembery told reporters at the Bahrain Grand Prix.
Hembery said he had tried to avoid setting a deadline for a deal, so as not to create unnecessary tension, but the world's fifth largest tyremaker "won't and cannot hang on forever.
"I've got a pretty good idea when that (deadline) is going to be and I've urged the teams to decide what they want very quickly," he said.
Formula One supremo Bernie Ecclestone told reporters that, from his part, a deal was signed but it also has to be agreed by the teams and the Paris-based governing International Automobile Federation (FIA).
Pirelli returned to the sport as sole supplier in 2011 following an FIA tender for a three-year period. In theory, the contract should again go out to tender although Hembery suggested the world body could simply ratify an extension.
The Briton said Pirelli had made a proposal but it was subject to regular review by the company's board.
He pointed out that European car sales had shrunk again in March after a dismal 2012, and that had a heavy impact on Pirelli although there was sales growth in many other markets.
"We are a European-based company and when we are taking tough decisions in Europe and you are looking to renew a significant project like F1, you have to keep going back to the board," he said.
"I had to present a document for our board again last month and it went back to them to review again the project. It's not my decision whether Pirelli stays in Formula One, it's the shareholders of the company.
"The sport we believe is a good platform for Pirelli going forward but big companies come under a lot of pressure. At a certain point someone has to make a decision somewhere," declared Hembery.
He made clear that Pirelli had plenty of other projects they wanted to do around the world and the resources would be reallocated if not needed in Formula One.
"We've made what we feel is a proposition that we can sustain with the current business environment and if that is acceptable then it's as it is. And if it's not, we need to know it's not," he said.
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