Questioned for several hours at the Munich hearing, Gribkowsky said the motoring magnate had offered him the multi-million dollar bribe in order to resolve a legal dispute linked to the sale of Formula One rights.
Gribkowsky was risk manager at BayernLB bank at the time, which was the biggest shareholder of Formula One but Bernie Ecclestone had a veto with a "golden share".
The German bank had decided to challenge that veto in court.
According to Gribkowsky's testimony, Ecclestone, in 2004, proposed during a telephone conversation to give Gribkowsky the money "to abandon the legal process".
Gribkowsky, on his first day of evidence, told the court that he informed his bosses and police about the offer, despite there never being an investigation into this latest allegation of bribery.
He also claimed that he was offered a second bribe of $80 million in Singapore, which appeared to take the court by surprise.
"And you are telling us this now, like this?" said Judge Peter Noll, who adjourned the hearing until next Tuesday.
Ecclestone, 83, is accused of paying Gribkowsky $44 million in 2006 and 2007 to ensure his continued grip on the motor sports empire he had built up over four decades.
Ecclestone, who has pleaded not guilty, faces a maximum jail term of 10 years if he is found guilty.
He admits making the $44 million payment not as a bribe but as hush money after a blackmail attempt.
Prosecutors charge that the money was a bribe meant to ensure BayernLB sold its shares to Ecclestone's preferred bidder, CVC Capital Partners of Britain, now the sport's majority shareholder.
The German was sentenced to eight-and-a-half years in prison in 2012 after being convicted of corruption in relation to the money.
It was the first day of testimony from Gribkowsky who is expected to give four days of evidence.