Mowatt and his representatives have been in talks with the Barnsley hierarchy over a new contract over recent months, but they are no closer to a breakthrough.
Back in May, Barnsley activated an option to extend Mowatt's contract until the summer of 2021, while also entering into talks over an extended deal.
Speaking to The Yorkshire Post, Murphy said: "We have been speaking to Alex and his representatives. We are still somewhat far apart in negotiations.
"But Alex is such a good person and his agent is a very good person that it will never get contentious and something that will become a headline.
"We are working towards something. At the end of the day, we cannot (currently) make ends meet.
"But Alex is our player for this season and our captain. He is vital to our success. The club wants Alex here regardless of what transpires over the next few weeks.
"We are continuing to work on something where both parties feel whole. From there, we just want Alex in our group."