Sky Sports Switzerland says Chelsea's owners, Todd Boehly and Behdad Eghbali, are losing confidence in Rosenior's ability to lead the club long-term.
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The board want to bring stability to the managerial position over the long-term, but do not see former Strasbourg coach Rosenior as having the qualities to do so.
As such, there is discussion inside Chelsea about removing Rosenior at the end of the season.
The news arrives after Chelsea lost 1-0 at home to Newcastle on Saturday, so leaving the club sitting in sixth place on the Premier League table.
Interestingly, along with Boehly and Eghbali, also part of the ownership team is Swiss medical equipment magnate Hansjörg Wyss, 90.