Craven made disparaging comments about Bach and two IOC members this month in a call he thought was with former Olympic track and field champion Edwin Moses.
The IPC chief, who is also an IOC member, had said "public opinion across the world is against (Bach)" over the IOC's decision not to ban all Russian athletes from the Rio de Janeiro Olympics due to a doping scandal.
The IOC had only suspended the country's track and field team from attending Rio over the fallout of a state-backed doping system in the country.
Craven, whose organisation suspended all Russian athletes from competing at the Paralympics in Rio because of the same doping affair, also had said German Chancellor Angela Merkel "couldn’t stand" Bach.
"The President has personally accepted the apology from Sir Phil about his comments," a source told Reuters, adding that Craven had written a letter to Bach.
"As far as Bach is concerned personally the matter is now closed."
In the recording of the prank call released by Russian television on Oct. 7 Craven also said he would "slam Bach" if he attempted to "pull a fast one" during an Olympic summit earlier in October to discuss doping.
"If Bach tries to pull a fast one, I'm going to slam him... and I hope Craig Reedie (President of the World Anti-Doping Agency) does the same.
"If Bach doesn't change his tune, he's going to be in a very difficult situation because public opinion all over the world is against him. And particularly in Germany. Germany can't stand him. Angela Merkel can't stand him."
Craven and Bach are due to meet in Tokyo during a World Economic Forum meeting on Thursday.
(Editing by Ed Osmond)
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