The men were in a BMW which crashed on Toller Lane, Bradford, in the early hours of Thursday morning.
The Independent Office for Police Conduct (IOPC) has begun an investigation following the crash, with a spokeswoman saying: "It is understood at this time that the car was being followed by an unmarked police car at the time of the crash."
A manager of a nearby coffee shop described those who died as "polite young men".
Mohammed Rashid, 51, said that the people he believed to have been involved in the crash had been going to his shop since they were boys and claimed they were all friends.
He added: "It's an absolute tragedy. Your heart just goes out to the families, to their parents.
"They were all polite young men, although I only knew them to greet and say hello to every now and then when they came in."
A large number of police and firefighters were called to the scene, near the junction with Bingley Road, at 5.30am.
The remains of the wrecked car were examined at the scene and then covered in a yellow sheet later on Thursday morning.
Nearby resident Roger Wood, 67, said: "It woke me up really early in the morning, there was this really loud bang down the bottom of the road.
"It wasn't until I woke up properly later in the morning that I found out what had happened, when one of my neighbours said they'd seen on the news that people had died. It's horrendous, horrendous."
Another nearby resident, who did not wish to be named, said that she had long-held concerns about safety on Toller Lane.
The mother of two said: "I was horrified when I first heard about it, but I wasn't surprised when I heard that it was Toller Lane where it had happened.
"It's known for being a bit of a speed-trap and there have been accidents down there in the past."
Christine Yates, who lives less than a minute's walk from the crash site, said that the car involved was "smashed up against a tree".
She added: "I knew that it was a very serious incident when I saw the number of emergency service vehicles that were there on the scene, as there were an awful lot of them."
Salim Rahman said that he too was woken by the sound of the car crashing into a tree.
The 69-year-old said: "I have been living on Toller Lane for 33 years and there are always cars coming up here too quickly. It's not unusual to hear sirens and police chasing people."
Describing the victims as "youngsters", he added: "It's incredibly sad to hear about it, your heart goes out to their parents."
IOPC regional director Miranda Biddle said: "This is a most tragic incident and our sympathies are with the family and friends of those who have died."
She said: "Our investigators are attending post-incident procedures, have travelled to the scene and have begun gathering evidence."