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The former Motherwell boss has emerged among the names linked with the Selhurst Park vacancy just months after leaving Scottish football for Ligue 1.
Askou spent a season at Fir Park before quitting for Toulouse in the summer, and his work in France has already caught the eye despite a mixed start in terms of results. His attacking philosophy has earned admirers, with the Dane now touted for a potential crack at the Premier League.
Palace are hunting for a new manager after pulling the trigger on Sage less than five months into his reign. Sunday's 3-1 home defeat to Liverpool left the Eagles rooted to the bottom of the table after just two wins and a draw from their opening 10 league games.
Former Tottenham and Brentford boss Thomas Frank is the leading candidate with the bookmakers, but Askou's name has also entered the conversation.
However, Palace have been told it is difficult to envisage Toulouse allowing the 43-year-old to walk away so soon after appointing him.
Théo Faugère, speaking to Bold.dk, admitted he was surprised to see Askou linked with the job but reckons his eye-catching style of football may explain the interest.
“I was a little surprised to see his name linked with Crystal Palace already. His attacking philosophy has probably attracted attention during his first months in Ligue 1.”
Results have not always matched performances for Toulouse, who were 12th after 10 rounds, but Faugère believes the early signs under the ex-Motherwell manager have been encouraging.
“Jens Berthel Askou's start has not been especially impressive when it comes to results, but the team's style of play has been interesting.
“From the first day, he made it clear that he wanted to introduce an attacking and aggressive style, and that was evident in Toulouse's first matches under him.
“In Ligue 1, the team's first home games against Lyon and especially Lille were very promising. Toulouse dominated large parts of both matches, but still ended up losing because they were not clinical enough in front of goal.
“Nevertheless, Toulouse create a lot of chances, and that makes me believe things will eventually fall into place.”
And that promise is why Faugère believes Palace would struggle to tempt Toulouse into giving Askou up midway through his first campaign.
“The club have not publicly commented on the interest. I find it very difficult to imagine Toulouse letting him go so early in the middle of the season.
“The club have repeatedly emphasised the importance of stability and had Carles Martinez Novell as coach for three consecutive seasons.
“Once again, the team under Askou are showing promising signs in terms of their football, so there is no reason why the club would not want to continue working with him.
“On top of that, it would be difficult to find a suitable replacement at this point in the season.”
Askou's emergence as a potential Premier League target represents a rapid rise following his spell in Scotland.
His immediate focus, though, remains on Toulouse and turning encouraging performances into more consistent Ligue 1 results, with a trip to Lens awaiting after the international break.
And while the Palace link is another indication of Askou's growing reputation, the message from France is clear: Toulouse have little appetite to abandon the project they began with the former Motherwell manager just a few months ago.