Clarkson, who hosted one of the most-watched television shows in the world, was dropped by the BBC Wednesday after an internal investigation
by the British broadcaster found he had subjected producer Oisin Tymon "to an unprovoked physical and verbal attack."
Tymon went to a hospital emergency department to receive treatment for a split lip following the March 4 incident at a hotel in North Yorkshire, it said.
North Yorkshire police said they had asked the BBC for a copy of the report and would assess whether further police action needed to be taken.
"No one who was present at the hotel during the incident came forward to report an offense to the police. Nonetheless, we have a duty to investigate where we believe an offense might have been committed, and that is what we have been doing with this case," police said in a statement
"As is usual in these circumstances, we have made contact with Mr Tymon through his lawyers, to ask him to speak to us so we can ascertain how he wishes to pursue this matter."
Tymon issued a statement through his lawyer Friday
saying he had informed police he did not want to press charges.
"The events of the last few weeks have been extremely unpleasant for everyone involved. The matter has taken a great toll on Oisin, his family and his friends," lawyer Paul Daniels said.
"Quite simply, Mr Tymon just wishes to return now to the job at the BBC he loves, as soon as possible. Further, the BBC have, in his view, taken action with a view to addressing the issues at hand.
"Mr Tymon agrees with the BBC's stated view that all parties should now be allowed to move on, so far as possible."
Tymon had earlier said
he was grateful to the BBC for their "thorough and swift investigation into this very regrettable incident."
"I've worked on Top Gear for almost a decade, a program I love. Over that time Jeremy and I had a positive and successful working relationship, making some landmark projects together. He is a unique talent and I am well aware that many will be sorry his involvement in the show should end in this way," he said.
Clarkson was suspended on March 10
after what the BBC initially described as the "fracas" with Tymon.
Ken MacQuarrie, who conducted the internal investigation into the incident, said Tymon had been struck "resulting in swelling and bleeding to his lip."
The physical attack was halted after about 30 seconds by the intervention of a witness, MacQuarrie said, but Clarkson continued to use "derogatory and abusive language" for a sustained period of time. MacQuarrie said Clarkson made a number of attempts to apologize over subsequent days and had reported the incident to BBC management.
"A member of staff -- who is a completely innocent party -- took himself to Accident and Emergency after a physical altercation accompanied by sustained and prolonged verbal abuse of an extreme nature. For me a line has been crossed," he said. "I know how popular the program is and I also know that this decision will divide opinion."
In 2013, Guinness World Records
named "Top Gear" the world's most widely watched factual program, with an estimated 350 million global viewers. The show -- fronted by Clarkson, James May and Richard Hammond -- is sold to 214 territories worldwide. Local versions have been made in the United States, China, Russia, Australia and South Korea.
After his suspension by the BBC, Clarkson changed his Twitter profile
to "I am probably a presenter on the BBC2 motoring show,Top Gear." On Wednesday, the wording became past tense: "I used to be a presenter on the BBC2 motoring show,Top Gear."
Co-host Richard Hammond
tweeted: "Gutted at such a sad end to an era. We're all three of us idiots in our different ways but it's been an incredible ride together."
On Friday James May also tweeted, saying: "I've written some blues lyrics: Oh I woke up this morning, And ... #Still Unemployed."
Fans had earlier expressed outrage at the BBC decision to suspend Clarkson.
An online petition
to have Clarkson reinstated has been signed by more than a million people. Fewer than 10,000 people signed a counterpetition calling for him to be fired.
Last week, a fan dressed as the "Stig" -- the anonymous racing car test-driver who was once a regular feature of the show -- drove to the BBC's London headquarters in an armored tank to present the "Bring Back Clarkson" petition.
Clarkson later thanked his supporters in a tweet shared more than 22,000 times.
While Clarkson's abrasive style has proven popular with viewers, his on- and off-air comments have earned him a reputation as a politically incorrect maverick who often walks a fine line between humor and offense.
Last year, he apologized profusely after being accused of mumbling the n-word in a clip that wasn't aired.
"I'd actually used the word I was trying to obscure. I was mortified by this, horrified. It is a word I loathe," Clarkson said in video statement posted online.
He's been accused on other occasions of racism, including characterizing Mexicans as "lazy and feckless" and using the word "slope" over footage of an Asian man crossing a bridge during a "Top Gear" special in Myanmar. Producer Andy Wilman later apologized, calling it a "light-hearted word play joke," and saying that the team was not aware that it was offensive to Asians.
Last week, Clarkson launched into an expletive-filled rant at a charity auction in north London, verbally attacking his BBC bosses. A CNN reporter who was at the event said Clarkson swore often as he talked about his suspension from the show, saying the BBC had "f**ked themselves" and had ruined a great show.
Clarkson later brushed off the incident, saying the rant was meant "in jest" and was designed to increase bids for the prize being auctioned -- one last lap of the "Top Gear" race track.