Celtic claim a fee was agreed with Manchester City for John Guidetti prior to them taking the striker on loan.
Xabi Alonso claims no Premier League offers were put to him before leaving Real Madrid for Bayern Munich.
Egervori's men were well placed to qualify for the World Cup play-offs prior to the clash, starting the night in second, but they ended it wounded and in fourth.
A passage to Brazil is now all-but closed off to them and 63-year-old Egervori's offer to step down was accepted by the Hungarian Football Federation.
In a statement on its official website, the federation said: "The resignation was accepted immediately."
Egervori added: "It is fair that I offered my resignation. It was a real blow for Hungarian football and it was appropriate.
"I hope the hard work of the last three years will not be lost."
Egervori's assistant, Joseph Csabi, will take the team for the final World Cup qualifier against Andorra.