When he left Selhurst Park in May, Allardyce said he had "no ambitions" to take another managerial job.
He continued: "Ultimately, it's the ambition of the club, the ambition of the owner and the board, allied to the fact that it's Everton and all the fantastic history, which have proved to be key for me".
The price of relegation is too much to consider for any established Premier League team these days.
When asked about Allardyce, Unsworth said: "Sam is a top manager, but I think it would be wrong of me to talk about individuals.I have seen the names that have been bandied about - all top managers".
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But former Arsenal and England striker Wright said there had been enough evidence for English football's governing body to act. United beat Brighton thanks to Lewis Dunk's own goal, which deflected in from Ashley Young's effort.
Palace chairman Steve Parish agreed to his departure and it is believed the south London club would receive a compensation package of around £2m if Allardyce takes a new job before next summer.
"I have no ambitions to take another job", he added at the time, "I simply want to be able to enjoy all the things you can not really enjoy with the 24/7 demands of managing any football club, let alone one in the Premier League".
Sam Allardyce will take charge for the first time in Saturday's Premier League clash at home to Huddersfield, having signed a deal to the end of June 2019.
While the return of Wayne Rooney cost the club nothing, the additions of Gylfi Sigurdsson, Sandro Ramirez, Jordan Pickford, Ashley Williams, Michael Keane and Davy Klaassen set them back in the region of $200 million.
His final game in charge was Wednesday's Premier League victory over West Ham at Goodison Park, which Allardyce watched from the stands.
Having spent heavy to bring in reinforcements over the summer, Everton have struggled to click this season; the Toffees now sit 17th in the Premier League, two points above the drop zone, and have already been dumped out of the Europa League, following a disastrous showing in the group stage.