The International Football Association Board (IFAB) met at Wembley on Wednesday to discuss the possibility of a trial, after a request from world players' union FIFPRO and the World Leagues Forum to test the protocol in the Premier League, France's Ligue 1 and Major League Soccer in the United States.
A trial of permanent concussion substitutes is to continue indefinitely.
But brain injury campaigners believe having the additional option of temporary concussion substitutes, as happens in rugby union, would provide players with the best protection by allowing medical staff more time to make an informed judgement.
"There are different points of view and they all have merit. I'm probably not the best person to give you both sides of the argument, but there won't be IFAB support for a trial right now," said Bullingham after seeing his proposal rejected.
Meanwhile, IFAB have approved a trial that will allow fans in the stadium to hear referees explain decisions taken after VAR reviews at next month's Club World Cup.
Bullingham said that while conversations between an on-field referee and the VAR official would remain private, the conclusion of their discussion would be relayed to the crowds at the event in Morocco.
"We think it's important in terms of transparency," said Bullingham.
"Predominantly to the crowd in the stadium, who at the moment don't get enough information as to what's happening with a decision."