Cops are investigating whether someone fishing off a Brooklyn promenade may have used their bait knife to slash the throat of jogger Katie Suarez last week, law-enforcement sources told The Post on Monday.

The theory surfaced as police explore every possible avenue in the baffling case, since investigators do not yet have a motive, much less a suspect, in what they currently believe is a random attack, sources said.

Detectives came up with the possibility since many people fish in the area of the attack and would have had a knife on them at the time for such things as cutting bait, sources said.

The startling attack came after a jogger was raped in Central Park last month, and police noted Monday that some details of Suarez’s stabbing echo the murder case of Karina Vetrano, a 30-year-old Queens woman whose body was discovered by her father after she was slain on a jog by a random attacker in 2016. That killer is now behind bars.

“Listen, it’s very eerily similar to the Howard Beach Jogger case,” Chief of Detectives Joseph Kenny told reporters of Suarez’s attack and Vetrano’s murder.

“It’s not too far from where it was,” he said of both incidents. “I ran that case as well. “I actually went back and read that case … to see if there’s anything jogs my memory that might assist in this.”

The revelations came as Suarez, who is brain dead, remained on life support because her family can’t bear to take her off yet, a relative told The Post.

“The mom is really hesitant on pulling the plug. She doesn’t want to see her daughter leave just yet,” said cousin Gianluca Frances, 19.

“It’s heartbreaking. Right now the defibrillator is currently what’s sustaining her to maintain oxygen in her lungs, so she’s not looking good,” Frances said of his 28-year-old cousin. “There’s some family that still wants to come to the hospital to see her.”

Family members told police that Suarez — who was stabbed on both sides of her neck as she jogged on the Shore Parkway Greenway promenade around 8 p.m. Friday — had no known boyfriend or enemies.

Detectives have been scouring her pink-cell-phone and computer records to see whether there might have been something her family did not know about, sources said.

Saurez, who graduated from Brooklyn College, had just gotten off work as a TD Bank teller at Caesar’s Bay Shopping Center in Gravesend and met up with her mom in front of a nearby Wendy’s restaurant around 6:50 p.m., sources said.

She left her mom to go jogging along the Belt Parkway, and the mother was soon joined by her dad — a typical ritual for the family, cops and sources said.

The dad, who was still with the mom at this point, called their daughter at 7:56 p.m. to see where and when they should meet up, again, as usual, police and sources said.

Suarez told them she was about 5 minutes away.

When she failed to meet up at the agreed-upon spot, the dad called her again — but Suarez didn’t pick up this time.

The parents then split up to look for her, and the dad ended up finding her stabbed on the path at 8:16 p.m., police said.

The critically wounded young woman had tried to continue to run after being attacked but only about 150 feet before she collapsed under a tree, police said Monday.

“After a period of time, the father begins to get worried because he knows she should be meeting them,” Kenny said.

“The father then begins to walk the path, and unfortunately, he finds her underneath a tree, bleeding heavily from the neck.”

The distraught father then called 911, along with another bystander who was walking on the path with his girlfriend, Kenny said.

Police were looking to talk to other walkers and joggers on the path as well as a food truck operator, who may have security footage and any drivers on the Belt that may have dashcam video, the chief said.

“In that 20-minute time period that we think that this attack could take place, over 1,700 cars drove by. So this is a well-traveled location,” Kenny said.

As a memorial grew at the family’s nearby home, workers at a Shore Parkway Wendy’s that Suarez frequented shared fond memories of their regular customer.

“She was very nice. She was an outgoing person. It’s very sad,” said a Wendy’s worker who confirmed she worked at nearby TD Bank. 

Saurez’s coworkers were tight-lipped, but the bank issued a statement saying it was “deeply saddened by this tragic incident involving a TD colleague” and was “actively cooperating with law enforcement.”

As police continued to explore every possible avenue for a break in the case, Suarez’s cousin called for additional resources to deter future attacks.

“Hopefully a story like this can encourage public policy handlers to incorporate policemen or officers into the park, or a camera installed,” Frances said.

Kenny said, “We have a tremendous amount of assets dedicated to this.

“We’ve set up a task force. We have major case on it. We’re really working it on. We’ve done we’ve done a lot of work in the last few days.”

Police meanwhile pounded the pavement in the area Monday handing out fliers offering a $10,000 cash reward for information leading to an arrest. K-9 units, scuba teams and drones also have been deployed to look for any leads.