The Eagles are having a disastrous start to the season under the Austrian boss, without a win in their first eight Premier League games, boasting just three points to their name and having scored only five goals.
After a strong end to the last campaign with Glasner in charge a lot was expected of Palace this season but it already seems likely that they will be battling relegation.
The south London club remain off the bottom of the table thanks to Wolves and Southampton going through even worse campaigns so far, with just one point each.
The Guardian report that Glasner must improve things before the next international break to keep hold of his job, giving him just four matches to turn things around.
These games see Palace host Tottenham on Sunday before travelling to Aston Villa in the Carabao Cup and then Wolves in the league before a home meeting with Fulham.
If Glasner has not picked up some wins then he could be out of Selhurst Park and the Guardian’s report states that Southgate could well be a target to replace him.
The former England manager started his professional career with the Eagles, making his debut for the club in 1990 and going on to play nearly 200 times for them before leaving for Aston Villa.
Southgate was a big hit as a player at Palace, becoming captain of the club at just 23 years old, playing in midfield before he was later moved into central defence.
‘I won’t coach in the next year, for sure,’ Southgate said at the ECA Europe general assembly. ‘I’m certain of that.
‘I need to give myself time to make good decisions. When you come out of a really big role you need to give your body time, you need to give your mind time.’
The 54-year-old is not the only name linked with replacing Glasner, with Graham Potter and David Moyes also touted as candidates.
Potter remains out of work since leaving Chelsea in April 2023, but did hold talks with Palace when they replaced Roy Hodgson with Glasner earlier this year.
Moyes left West Ham at the end of last season and the 61-year-old has been regularly linked with Premier League jobs since, including former club Everton and struggling Southampton.
If Glasner is to lose his job then it would be a remarkable deterioration for him after a fabulous start at Palace and being linked with the Bayern Munich job over the summer.
Bayern Munich honorary president Uli Hoeness revealed that the club had spoken to Glasner about the manager’s role which Vincent Kompany eventually took up, but Palace had no intention of letting him go.
‘We spoke with Glasner, who we also liked,’ Hoeness said. ‘He was very keen on joining FC Bayern, absolutely keen. But his club did not release him, and FC Bayern is not all-powerful either.’
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