Weah senior took in a loan spell at Stamford Bridge back in 2000, netting three times in 11 appearances.
The 1995 World Player of the Year has remained in contact with officials in west London and has arranged for his son to head for England.
Timothy is only 13 years of age, but he will be given an opportunity to impress Chelsea's coaching staff and earn himself a place in their academy ranks.
Weah told talkSPORT: "He is talented.
"We just wanted to give him the opportunity and we also want him to enjoy the game he has passion for, so I brought him to Chelsea just for a trial and for a different experience.
"When I left London, I kept ties with Chelsea and when my son wanted to come to England to play, the first person I called was the club's liaison officer Gary Staker.
"Gary made it happen and Timothy is enjoying it. He has started training already and he is happy."