Pierce County sheriff's spokesman Ed Troyer told reporters that Maurice Clemmons, 37, was one of several people investigators want to talk to and that he could not be called a suspect at this point.
In a news release, the sheriff's office said Clemmons has an extensive violent criminal history from Arkansas, including aggravated robbery and theft. Clemmons also recently was arrested and charged in Pierce County in Washington state for third-degree assault on a police officer, and second-degree rape of a child.
The four officers were with the 100-member police department of Lakewood, which adjoins the unincorporated area of Parkland, where the shootings took place. The city identified the victims as Sgt. Mark Renninger, 39; Ronald Owens, 37; Tina Griswold, 40; and Greg Richards 42.
Troyer said one of those officers fought with the gunman and may have wounded him before the officer died just outside the doorway. He told reporters that investigators were asking area medical providers to report any people wounded by gunshots.
Troyer said investigators believe two of the officers were shot dead while sitting in the shop, and a third was killed after standing up. The fourth apparently struggled with the gunman out the doorway and "gave up a good fight," getting off a few shots before he was either shot there or succumbed to earlier wounds.
"We believe there was a struggle, a commotion, a fight ... that he fought the guy all the way out the door," Troyer said.
He added, "We hope that he hit him."
Troyer said the gunman entered the coffee house and walked up to the counter as if to place an order. A barista saw a gun when the man opened his jacket and fled out the back door. The man then turned and opened fire on the officers as they sat working on their laptops, killing the three men and one woman in what Troyer described as a targeted ambush.
Troyer said the attack was clearly targeted at the officers, not a robbery gone bad.
"This was more of an execution. Walk in with the specific mindset to shoot police officers," he said.
Troyer said the officers - all from the Lakewood Police Department - were catching up on paperwork at the beginning of their shifts when they were attacked at 8:15 a.m. Sunday.
"There were marked patrol cars outside and they were all in uniform," Troyer said.
There was no indication of any connection with the Halloween night shooting of a Seattle police officer. The suspect in that shooting remains hospitalized.
"We won't know if it's a copycat effect or what it was until we get the case solved," Troyer said. "We don't even have a suspect ID right now."
Troyer estimated that a couple of hundred officers from the Washington State Patrol and multiple surrounding police agencies in the area were at the crime scene, with some coming on their own time.
"We have no motive at all," Troyer said. "I don't think when we find out what it is, it will be anything that makes any sense or be worth it."
Two employees and a few other customers were in the shop during the attack. All were interviewed by the Pierce County sheriff's investigators.
"Some are in shock. They are very upset," Troyer said. "They are the ones who are going to put together for us how this happened."
The Forza Coffee Shop, part of a popular local chain, is on a side street near McChord Air Force Base in Tacoma, about 35 miles south of Seattle. The shop is in a small retail center alongside two restaurants, a cigar store and a nail salon.
Brad Carpenter, founder and owner of Forza Coffee, said his staff was OK and being interviewed by police, and that his main concern was for the families of the police officers.
"I'm a retired police officer, so this really hits close to home for me," said Carpenter, of nearby Gig Harbor.
Troyer said the Lakewood officers were two blocks outside their jurisdiction, and the coffee shop was a popular place for officers from surrounding jurisdictions to meet and share information.
Streets around the coffee shop were blocked off late Sunday morning, and a police helicopter hovered over a large crowd of investigators. TV video showed police taking possession of a pickup truck parked in a grocery store in Parkland.
Troyer said investigators were checking surveillance video from multiple sources, trying to identify a possible getaway car.
Dave Gabrielson, a clerk at Foot Mart about a block away from the coffee shop, told the newspaper all was quiet when he opened the store at 8 a.m. About 30 minutes later, "All of a sudden a million cops were zooming up and down the road," Gabrielson said.
He said he saw officers bring a police dog into a nearby apartment complex.
This is the second targeted police shooting in the area in the last month, reports CBS News correspondent Hattie Kauffman. Seattle police officer Timothy Brenton was shot and killed Halloween night as he was sitting in a cruiser with trainee Britt Sweeney. Sweeney was grazed in the neck.
Authorities say the man charged with that shooting also firebombed four police vehicles in October as part of a "one-man war" against law enforcement. Christopher Monfort, 41, was arrested after being wounded in a firefight with police days after the Seattle shooting. He remains hospitalized in stable condition, the hospital said Sunday.
The officers killed Sunday were a patrol squad made up of three officers and their sergeant. No threats had been made against them or other officers in the region, sheriff's officials said. Their families have been notified.
"We lost people we care about. We're working to find out who did this and deal with him." Pierce County Sheriff Paul Pastor told reporters at the scene.
Washington Gov. Chris Gregoire said she was "shocked and horrified" by the killings.
"Our police put their lives on the line every day, and tragedies like this remind us of the risks they continually take to keep our communities safe," she said in a written statement. "My heart goes out to the family, friends and co-workers of these officers, as well as the entire law enforcement community."