Teams discussed the failure of the new elimination format with F1 commercial boss Bernie Ecclestone and Jean Todt, president of governing body, the FIA.
Mercedes F1 boss Toto Wolff said Todt and Ecclestone refused to revert to the 2015 system despite teams' wishes to do so.
A new aggregate system proposed by the FIA will be discussed again next week.
Lewis Hamilton qualified fastest for Sunday's race in Bahrain, but there were quiet periods near the end of the first and second sessions when there were no cars out on the track.
Red Bull team principal Christian Horner said all parties had agreed that the new system introduced on the eve of the 2016 season - where the slowest car is eliminated every 90 seconds in three sessions - was not the way forward.
Horner said: "There is an unwillingness from the promoter and FIA to go back to 2015. The teams would go back. A compromise has been put on the table now for the teams to consider.
"Let's have a look at what's been out on the table today. The bottom line is if we don't agree to a compromise, then we're stuck with what we've got and everybody agrees that what we've got isn't right."
Horner and Wolff both said they did not know why Ecclestone and Todt were so opposed to reverting to the 2015 system, whereby all eligible cars ran to the end of a session before the slowest few in each are knocked out, gradually reducing the number of cars over the three sessions.
Todt said in a news conference on Saturday that race promoters had asked for a different system because they feared the 2015 approach was not exciting enough.
Ecclestone has made it clear he wants to find a way to shuffle the grids so that the fastest cars are not always at the front.
Horner said that the proposal was for a reversion to the 2015 format, but with each drivers' two fastest laps being counted towards their grid position rather than just their fastest.
Horner said the plan "needed a bit more investigation, perhaps with an aggregate time of two laps rather than a single lap. So it needs to be properly thought through and considered before voting on it."
Formula 1 bosses have failed to agree on a new format for qualifying after a meeting at the Bahrain Grand Prix.