The German coach has arrived on Tyneside as Eddie Howe's replacement with the club preparing to sell captain Bruno Guimaraes to Arsenal on the back of already offloading Anthony Gordon and Sandro Tonali earlier this summer.
Jaissle will sit down with several first team players in the coming weeks with more outgoings expected before the end of August.
Murphy was a highly valued squad star under Howe, but as he's now into the final year of his contract, the club want a decision over a renewal from the 31-year-old winger.
He agreed fresh terms last summer, but that involved a wage increase, rather than extending his deal beyond 2027 and Football Insider claim Everton will make an offer if he asks to leave.
The club hierarchy will ask Jaissle to make a call on Murphy's place in his plans as quickly as possible, as this summer is their final realistic chance to land a fee for the former Norwich City man, and the Toffees could open with a £5M bid.