Police spokeswoman Lindsey Eldridge said the deputy was shot by Green, who was killed after the deputy returned fire.
The deputy was taken to University of Maryland R Adams Cowley Shock Trauma Center in Baltimore. Officials did not identify the marshal.
As of Thursday afternoon, physician in chief Thomas Scalea said the deputy remained in the intensive care unit on life support following surgery.
“We are very hopeful, but you just never know,” Scalea said. “It’s just too early.”
The deputy is a member of a Special Operations group and a veteran of the U.S. Marine Corps, said Don Snider, commander of the U.S. Marshals Capital Area Regional Fugitive Task Force.
Snider said the shooting occurred when marshals were clearing the residence of an address associated with Green and a man began firing shots from a closet.
The officers returned fire, fatally shooting the man in the closet who was later identified as Green.
“I’d really like this to be the last time I have to walk up here and do this, but that’s not likely to happen,” Scalea said of briefing the press on the marshal’s condition. “It’s really — it’s a little bit too much.”
The Baltimore Police Department will conduct the investigation into the shooting.
Baltimore Police Commissioner Michael Harrison said Green was wanted on 19 charges, including armed robbery and attempted murder.
On Saturday, Green opened fire at Baltimore police officers at a grocery store in northern Baltimore before fleeing, Harrison said.
“It goes without saying,” Harrison said, “going after violent offenders is one of the most dangerous jobs in America.”
The Baltimore Sun reported that the man who was shot Thursday had been a security guard at the grocery store and was unhappy with his paycheck.
“The entire city of Baltimore’s thoughts and prayers are with the deputy marshal,” said Baltimore Mayor Brandon Scott (D). “This is a very dangerous situation, and we have to remember that we will be going after these violent criminals.”
Family for Green could not be immediately reached.