Club owner Ashley was told he would receive no support from Newcastle City Council for his decision to rename St James' Park.
On Friday the council wrote to news organisations asking for their support.
Labour and Liberal Democrat councillors united to pass a motion opposing the decision after the name change was announced in November.
They called on the sportswear tycoon to reconsider, and refused to change street signs in the city from St James' Park to Sports Direct Arena.
The motion, passed at a meeting of the full council, also committed city officers to write to media organisations asking for their support.
The letter said: "The motion requests that the city council write to the media and asks that they continue to use the name of St James' Park in all reporting and refuse to use the name Sports Direct Arena.
"We respectfully request that you take this approach.
"For its part, the city council has taken the decision not to change any wayfinding signs which bear the name St James' Park, and calls on the club to reconsider their decision.
"We would ask that you acknowledge the strength of opposition among Newcastle fans against changing the name of the stadium and continue to call it St James' Park in all reporting."
Newcastle United refused to comment.
St James' Park has been the home of Newcastle since 1892.
The stadium will be called the Sports Direct Arena until a new global sponsor can be found, in a move club bosses say will bring money to the club.
Derek Llambias, the club's managing director, said it could generate up to £10 million a year.
The BBC said in a statement: "As of 10 November, the Sports Direct Arena became the official name for the ground as chosen by the club.
"It is not the BBC's place, as an impartial public broadcaster to decide or debate on whether this is correct or not, rather to report on the matches which take place on the field.
"Changing the way we display the graphics or refer to the ground on our programmes ensures consistency with every other ground which has a sponsor, be it new grounds that came with a sponsor like the Emirates, or old grounds that have changed their names like Bradford City's Coral Windows Stadium (formerly Valley Parade)."