The influential midfielder is contracted with the Reds until the summer, although there is an option of a further year in the club’s favour.
The Oakwell outfit have been in negotiations with Mowatt, 25, about an extended deal for a spell of time and are striving to make a breakthrough.
Murphy told The Yorkshire Post: “The idea is to keep our core group together. We believe we are going to stay in the Championship.
“But if we don’t, we would still like to keep the group who are keeping us successful on the field to remain at Oakwell.
“Alex is the only one right now who has only got one year left. Everyone else is on multiple-year contracts.
“We have had several extension talks and given several extension offers and have one on the table right now which we are working towards to get Alex on a longer-term contract,” added Murphy.
Barnsley recorded losses of £3.4m for the year ending May 31, 2019 – according to their latest set of financial results.
The loss compared with a modest deficit of £0.2m in the previous year’s accounts. The club’s latest Companies House financial update also revealed that turnover dropped to £7.8m from £14m in 2017-18.
Money spent on wages was £8.1m – compared to £10.6m in the previous year – while the club announced a profit on player trading of £1.3m in comparison to £1.2m in 17-18.
The club’s income from Football League distributions in 2018-19 – when the Reds were in League One – stood at £2.3m, down from the figure in the previous year of £7.9m when the club were in the Championship.