U.S. President Donald Trump on Tuesday said he wants a "full investigation" in the apparent suicide of alleged sex offender Jeffrey Epstein at a federal jail in New York City.

"I want a full investigation, and that's what I absolutely am demanding. That's what our attorney general, our great attorney general is doing. He's doing a full investigation," Trump told reporters in New Jersey.

The president's remarks came one day after U.S. Attorney General William Barr said the jail where Epstein died had "serious irregularities."

Barr said on Monday that initial discoveries made by investigators probing into the death of Epstein found the jail is partially responsible for the incident, vowing to "get to the bottom of what happened" and promised accountability.

Barr also said the Justice Department would continue investigation in the possible accomplices of Epstein in his alleged sex crimes.

Epstein was found unconscious in his cell on Saturday and was later pronounced dead.

U.S. media reports said the jail had not followed protocol to secure Epstein's safety prior to the apparent suicide. Epstein was at the time under "special observation status," indicating that he should have had a cell mate and jail guards should have checked on him every 30 minutes. But records showed that the cell mate was not present when the apparent suicide took place and jail guards failed to make the scheduled tours.

Epstein was accused in early July of sexually exploiting and abusing dozens of underage girls in the early 2000s. He was denied bail weeks after and would face up to 45 years in prison if convicted.

After a previous attempt to take his own life, Epstein was placed on suicide watch, but was later removed from suicide watch at the request of his attorney.