The three outfits lobbied against Pirelli's push to tweak specifications for the Canadian GP because they feared the changes to cure delaminations would affect the competitive order.
But after a number of spectacular blow-outs at Silverstone, the teams say that matters have now got more serious and that safety concerns need addressing.
Force India deputy team principal Bob Fernley, whose outfit was most strongly against the revisions last time, told AUTOSPORT: "It's a completely different position to what we had previously because there was no safety element in any of it before.
"Once a safety issue comes up, we are not going to risk people's well-being for a technical point."
Stefano Domenicali, Ferrari team principal, said: "For me, it's important to move forward and solve the issue because it's something that is very important for all of us."
Lotus boss Eric Boullier added: "We need to work with Pirelli and find solutions.
"Maybe Pirelli need to change the belt from metallic to Kevlar, and we would support this. Safety is the primary concern."
Force India team principal Vijay Mallya has suggested that, while his team will not block it, he is not totally in favour of any tyre adjustment.
"We are having no issue, so why should we make other people's issues our issues?" he said.
"But there is no question of this team being irresponsible in any way, and objecting for the sake of objecting to any proposed change that impacts on safety in Formula 1 - no question.
"However, if we believe that a few teams have understood the tyres better, and are able to manage the tyres better, without compromising safety, why should we vote for change?"
Mallya also reckoned that the safety issue was being over-emphasised because of the spectacular way the tyres came apart.
"Other than the car behind and the driver who might get a bit startled with the sudden explosion in front, it really isn't a major safety issue from that point of view," he said.
"We didn't see cars wildly spinning off and crashing into the barriers.
"But Pirelli needs to do what it needs to do and work with the various teams to come up with an appropriate solution."