The force is also looking to award him with a commendation from Commissioner Ian Dyson.
As the plans were announced, Mr Olubunmi-Adewole's family described him as a "loving son" and "best friend" who is "irreplaceable".
Mr Olubunmi-Adewole was on his way home from work at a central London restaurant when he entered the water about midnight last Saturday after spotting a woman fall from London Bridge.
A City of London Police spokesman said: "We feel that is absolutely right to recognise the bravery and selfless actions of Folajimi Olubunmi-Adewole, known as Jimi.
"To honour his memory and heroism, we have contacted the Royal Humane Society to start the nomination process for their bravery award, and we are also looking to award Jimi a commissioner's commendation from the City of London Police.
"Our thoughts remain with his friends and family at this difficult time."
In statements through the force, his father Michael Adewola said: "Jimi is a loving son and my best friend. He has a good heart and always willing to help everyone around him. I'll forever miss him."
His mother Olasunkanmi Adewole said: "My son Jimi is my strength, he takes care of me and everyone. I'm empty without Jimi. I pray to have peace in my life."
His brother described him as "irreplaceable" and "the most generous person".
City of London Police previously said two men entered the water after seeing a woman fall from the bridge last Saturday.
The coastguard and the Metropolitan Police marine unit rescued the woman and one man, but could not locate the other man.
A body was found at 6am on Saturday morning, police said, which was later formally identified as Mr Olubunmi-Adewole.