Lawyers acting for the Dudley Group NHS Foundation Trust, which is charged with two breaches of the 2008 Health and Social Care Act, did not enter any plea during the 30 minute hearing at Dudley Magistrates Court on Wednesday.
The trust is being prosecuted after a Care Quality Commission (CQC) investigation into the deaths.
The health and social care regulator has said the allegations relate to healthcare providers' responsibility to ensure people receive safe care and treatment.
In a statement issued on Tuesday, the CQC said it had brought the prosecution following two specific incidents in which patients died after being allegedly exposed to "significant risk of avoidable harm" at Russells Hall Hospital, in Dudley.
One of the patients who died was Natalie Billingham, from Tipton.
The 33-year-old mother-of-six died from sepsis in March 2018, after she was originally admitted into A&E with suspected deep vein thrombosis.
The CQC has also confirmed the second patient involved was Kaysie-Jane Robinson, from Netherton.
During Wednesday's initial hearing District Judge Graham Wilkinson told the trust's representatives they would be required to enter pleas on July 2.
Counsel for the CQC, Ian Bridge, told the court the charges related to clinical care and governance, and that both deaths related to patients being treated for sepsis.
"The families (of the patients) are aware of today's hearing but, anticipating that it would be administrative, we have not arranged for them to attend," Mr Bridge said.
In a statement provided to the Express & Star on Tuesday, a Dudley Group NHS Foundation Trust spokesman said: "It would be inappropriate for the Trust to comment on the CQC prosecution while court proceedings are active.
"We do, however, want to reassure our patients and the public that the hospital provides a safe and compassionate environment for their care and that we continue to build upon the service improvements we have made since these two cases in 2018."