Folajimi Olubunmi-Adewole, known as Jimi, was on his way home from work at a central London restaurant late on Saturday when he was alerted to a woman falling from London Bridge.
His friend Bernard Kosia, who was with him at the time, said they were approached by two men who showed them a video of a woman “jumping over the bridge”.
He said they called 999 and heard a woman shouting “I'm dying, I'm dying” but could not see anything because it was “pitch black”.
Mr Kosia said he could not swim but Mr Olubunmi-Adewole and another man went into the water at about midnight to try to rescue the woman.
He said he heard his friend shouting his name “but the shout wasn’t normal, it was pain”.
Describing Mr Olubunmi-Adewole as a “hero”, Mr Kosia told ITV’s Good Morning Britain: “Jimi was adamant, for some reason, that he didn’t care [about himself]. He didn’t think twice as to what was going to happen.
“I just remember him saying, ‘Bernard, I have to save her, I’m going to save her’.”
He added: “Jimi means a lot to everyone. Especially around south London now, everyone knows him as a neighbourhood hero.
“To risk your life in such a way, you don’t see that every day.
“And the fact that he was adamant to save that woman, it just shows his nature and his mindset.”
Recalling their last conversations before the incident, Mr Kosia said his friend had spoken about his family and told him he loved him.
He said: “They [our conversations] were just always about family, loved ones, what we have to do for them to make sure that they’re OK.”
Mr Olubunmi-Adewole’s parents, Michael Adewola and Olasunkanmi Adewole, wept as they paid tribute to their son.
Mr Adewola called for his son to be awarded a posthumous George Cross for bravery.
He told Good Morning Britain: “I want the government to honour him because Jimi was a good soul to me.
“I want the government to do something … so that his name cannot vanish forever.”
Describing his son as “the breadwinner of the house”, he said he was “always taking care of me and my wife and his friends”.
He thanked all those who have supported a GoFundMe page set up to help the family, which has so far raised more than £88,000.
Ms Adewole said her son “had a good heart”.
The couple have previously been reported as saying they believe the length of the search for their son in the water was “not enough” and that they “want justice” for him.
The coastguard and the Metropolitan Police’s marine unit rescued the woman and the other man, but could not locate Jimi.
A body was found at 6am on Saturday morning, police said, which has since been formally identified as that of Mr Olubunmi-Adewole.
Detective Chief Superintendent Oliver Shaw, from City of London Police, described him as a “brave, kind and selfless young man”.
He said: “Our officers arrived extremely quickly to the scene but two members of the public had already entered the water.
“One man had managed to swim to the woman and they were both seen above water by the coastguard. Sadly, there was no sign of the other rescuer.
“We remained on scene to assist the marine units and police helicopter with the search in the water. This was stood down by the coastguard over an hour later.
“We continued to search the shore of the river, in person and using our network of CCTV cameras, and, unfortunately, at 5.46am a body was discovered.
“The circumstances of this incident are extremely tragic. We continue to support the family of this heroic young man who heartbreakingly lost his own life trying to save that of another.”
A spokesperson for the coastguard said: “HM Coastguard's first priority is always to save lives and we will continue to search the surface of the water within a search area until either the missing person has been found or a decision is taken to stand down the search.
“These decisions are taken after serious consideration of all the facts and take into consideration the feelings of the family and friends of those who are missing.
“The search involved five search-and-rescue boats, visual searches from bridges and the police helicopter searching overhead. Despite initial conflicting reports about how many people had gone into the water, we based our search on the fact there could be three.
“It was determined where in the water the person could be based on tidal conditions and that area was thoroughly searched many times before the decision was taken to suspend the search at 1.20am.”
Additional reporting by PA
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