The Reds hierarchy have been speaking to the 26-year-old - out of contract at the end of this month and able to speak to other clubs - about a new deal for some time, but without making a breakthrough.
Championship Team of the Year member Mowatt - who back in January admitted to facing a ‘massive decision’ over his future will speak with the club on his return, with discussions temporarily placed on hold during the club's push for promotion at the end of the season.
Barnsley CEO Murphy still retains confidence that Mowatt - subject of a failed bid by a rival club at the start of the new year - will make the decision to stay put, in the final analysis.
The club fervently believe that Barnsley is the best place for the Doncaster-born player, whose career has been taken onto the next level by much-admired head coach Valerien Ismael
Ismael has a strong working relationship with Mowatt and has spoken to the player about continuing to progress his career under him at Oakwell following the best season of his career.
On Mowatt, who has been linked to several clubs, Murphy told The Yorkshire Post: “We have tried for close to a year - maybe even longer since I have been here - to extend him (his deal). We are still confident we can get that across the line.
“That being said, the core of our team is here to stay and we just have to make sure we are continuing to strengthen in the summer and make sure we take the next steps to compete at the highest level.
“With Alex, it is finding the best situation for him. I am strong in the belief - and I think Valerien is strong in the belief - that Barnsley Football Club is the best place for Alex.
“He is the captain and leads the line and is going to play and no-one is going to take the shirt off him.
“He is able to play in a role where he can get forward and have an impact to score goals and provide assists and impact the team defensively. I don’t know if he has a better (other) situation out there for him.
“He is on holiday right now and we are continuing to stay in communication and I think that when he gets back, we will sit down with him and his agent and try to figure out what the best steps forward are.”
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