The struggling refinery, operated by Murphy's British subsidiary Murco, this month began 45 days of consultation with employees on the future of the plant after exclusive talks to sell it to a private equity firm collapsed.
A Murco spokesman confirmed it had stopped purchasing crude oil for the plant, adding that the company was engaged in talks for its sale to unnamed parties.
"We are actively engaged with certain third parties that have expressed an interest in acquiring the refinery. These third parties are aware that Murco's crude purchasing has now ceased. We will be providing further updates in due course," he said in a statement.
The company had hoped to sell the 135,000-barrel-per-day (bpd)
Murco has now decided to halt crude oil processing at the plant in mid-May as no new buyers have come forward, the sources said.
"The plant continues to look for potential buyers but it will stop processing (crude) in mid-May," a source said.
The future of the plant and its 450 employees remains uncertain as official consultations talks have yet to be concluded, one source said.
A spokesman for Murco declined to comment. Murco will need to complete the consultation period before laying off staff.
Even if the refinery shuts down, the site could be converted to a storage terminal which could make use of deep-water berths at the site.
European refiners have struggled with shrinking domestic demand and increased overseas competition which has crushed margins and led to loss-making.
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