The Eagles boss was scheduled to speak to the media at 1:30pm amid reports he will be sacked.
But Hodgson will not be able to attend after being taken ill during Palace's training session this morning.
Posting on social media, Crystal Palace wrote: "Unfortunately, today's press conference will no longer take place as scheduled as Roy Hodgson was taken ill during this morning's training session."
Hodgson is on the brink of the sack at Selhurst Park after a 3-1 defeat to Chelsea on Monday night.
Oliver Glasner has been lined up to replace him, talkSPORT understands, with talks taking place on Thursday.
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Palace are set to face Everton on Monday with Hodgson set to field questions ahead of the Premier League clash, which could now be seen as a relegation six-pointer.
Hodgson was expected to be asked questions over his position as manager but the press conference was cancelled at just over half an hour's notice.
Palace chairman Steve Parish was believed to have decided on Hodgson's future after the defeat to Chelsea which left the Eagles with four wins 18 league games.
There are real fears that Palace could be sucked into a relegation battle this season with the club 15th in the table.
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Just five points separate them from the drop zone with Everton in 18th place, meaning Monday's match could be pivotal at Goodison Park.
Glasner has verbally agreed to become the new Palace boss, talkSPORT understands.
The Austrian coach left Eintracht Frankfurt at the end of last season and is extremely well thought of.
Glasner led Frankfurt to glory in the Europa League in 2022, defeating West Ham in the semi-finals before beating Rangers in the final.
Palace had tried to convince Kieran McKenna to leave Ipswich Town midway through the season but talks failed.
Hodgson was brought back to Palace for a second time last March to replace Patrick Vieira and successfully steered them away from relegation.
The former England boss, who managed the south London side from 2017 to 2021 as well, was only appointed until the end of the season but his revitalisation of the club saw him stay on for another year.
Palace had wanted to honour his contract and let him leave in the summer but the decision has been made to let him go now to avoid falling into further trouble.