An online petition calling for the BBC to reinstate Jeremy Clarkson has been signed by more than 150,000 people.
The Top Gear presenter was suspended and the show put on hold after he allegedly tried to hit one of the producers.
A spokeswoman for the BBC said no one else was suspended over the alleged "fracas" after filming in Newcastle.
The producer involved in the alleged row, said to have been over a lack of catering, has been named by the Daily Mirror as Oisin Tymon (36).
A BBC spokeswoman said: "Following a fracas with a BBC producer, Jeremy Clarkson has been suspended pending an investigation."
"No-one else has been suspended. Top Gear will not be broadcast this Sunday. The BBC will be making no further comment at this time."
The online campaign on Change.org was created with the social-media hashtag #BringBackClarkson by right-wing blogger Guido Fawkes.
The presenter's daughter Em Clarkson also tweeted her support: "Oh God, BBC please take him back... he's started cooking...".
Hours after the suspension was announced, Mr Clarkson's co-presenter James May also took to Twitter: "No Top Gear this weekend, apparently. How about [the film] 633 Squadron instead?"
Fellow host Richard Hammond replied: "No, surely, Last of the Summer Wine; no one will notice the difference. Job done."
Mr Clarkson, who has been involved in a number of high-profile gaffes in recent years, was put on what the corporation called his final warning in 2014.
It followed claims he used the n-word while reciting the nursery rhyme Eeny, Meeny, Miny Moe during filming of the BBC2 programme.
Mr Clarkson said he was "horrified" that it sounded as though he used racist language in the out-take, which was not broadcast.
The Sun newspaper, for which he writes a column, said he shrugged off his suspension and quoted him as saying: "I'm having a nice cold pint and waiting for this to blow over."
Top Gear is one of the BBC's biggest money spinners, pulling in millions of pounds from a devoted - and international - audience.
The latest series was given a global launch with a simultaneous broadcast in more than 50 countries.
This Sunday's episode was set to feature Mr Clarkson with co-hosts Hammond and May getting to grips with classic cars such as a Fiat 124 Spider, an MGB GT and a Peugeot 304 cabriolet.
They were going to take to the road and end up at a classic track day, with Gary Lineker due to feature as the "star in a reasonably priced car".
Former Top Gear presenter Chris Goffey thinks it will take a brave man to fire him.
