Friday's home defeat leaves Serbia second bottom in Group I with one point, having been docked three points by UEFA for their wrongdoing in the abandoned fixture with Albania last month.
With Denmark and Portugal six and five points better off respectively in the pool, Serbia's chances of reaching the finals in France look in grave danger.
As part of their UEFA punishment, Serbia were forced to play Friday's clash in front of an empty stadium in Belgrade, and Advocaat lamented the team's current fortunes.
"If you want someone to blame - blame me," he said.
"I sent them to the ground and told them what to play. I'll think about resigning.
"The problems started after Albania, after all these decisions and games without spectators. It's hard to win points in this atmosphere.
"I took the job with the Serbian national team for a reason. I believed that we have a team for the European Championships, but just a few months later the UEFA decisions put us under enormous pressure.
"The fair and proper thing is to sit down and talk with the leaders of the federation. I'll talk to them before the match with Greece [on Tuesday].
"I never in my career had a worse start in the qualification, however, remember what happened after Albania. Something is broken, it's a disastrous start."