Clarkson was sacked by the corporation on Wednesday after an internal inquiry concluded that he had punched Tymon in the face, leaving him needed medical assistance, following a row over catering arrangements.
The BBC announced it was liaising with police over what it described as an "unprovoked physical attack" after director general Lord Tony Hall's statement was released.
Lord Hall said that a "line has been crossed" with this latest incident, which involved a member of staff (Tymon) taking himself to A&E after a "physical altercation accompanied by sustained and prolonged verbal abuse of an extreme nature".
The BBC said North Yorkshire police asked to see the report into the incident.
A spokesperson for North Yorkshire Police said: "North Yorkshire Police is liaising with the BBC regarding the alleged incident in North Yorkshire involving Jeremy Clarkson.
"We have asked the BBC for the report which details the findings of their internal investigation into the matter.
"The information will be assessed appropriately and action will be taken by North Yorkshire Police where necessary.
"It would not be appropriate for North Yorkshire Police to comment further at this time."
However, Tymon has since decided not to take the complaint any further.
On Thursday, Clarkson called on his fans not to blame the producer, who had been trolled on social media, for the sacking.
Speaking to reporters outside his London residence, he said: "All I would like to say is I wish people would leave Ois alone, because none of this is his fault."
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