Mercedes' Toto Wolff, and Red Bull's Dr Helmut Marko, suggested in Malaysia that the sport's Italian supplier has agreed to bring modified tyres to Bahrain late next month.
But we reported on Wednesday that Lotus and Ferrari - both much happier with Pirelli's current tyres - would 'veto' any changes on the grounds that they do not pose a safety risk.
Ferrari's Stefano Domenicali was quoted by Auto Motor und Sport: "It makes no sense to panic after two races."
Paul Hembery, Pirelli's motor sport director, has also now ruled out making changes as the result of Mercedes and Red Bull's complaints.
"If we did," he is quoted by Spain's AS newspaper, "and we ended up favouring one team, we could have the championship over by Monza."
The entire tyre saga has quietened in the past days, perhaps because Red Bull and Mercedes monopolised the highest four finishing places in Malaysia.
"They (Red Bull) have good performance," said Hembery, "perhaps they would like to have more, but I'm sure that's the same for other teams too."
Domenicali agrees that "some teams have overreacted about the tyres".
Hembery continued: "This is F1; everyone trying to gain advantage over the others. If you have the whole paddock against you, then you have to take action."
The timing of the controversy is slightly uncomfortable for Pirelli, given its negotiations with Bernie Ecclestone and the FIA about remaining F1's sole tyre supplier beyond 2013.
Hembery said he hopes to receive news on this front "soon".