The Wellington Street offices were home to the Yorkshire Post and Yorkshire Evening Post for more than 40 years.
The tower displaying the time and temperature is a familiar sight for commuters driving into the city on the inner ring road.
Staff were moved out of the building last year and demolition has begun.
Paul Fox, of agent Fox Lloyd Jones, said: "The developers have noted the affection people have for the tower and wish to retain it as a landmark."
Mr Fox added the public had a "love/hate relationship" with the building but the tower would provide a "lasting memory".
It is planned to re-clad the tower at a cost of Â£250,000 and for it to display digital marketing, he said.
A planning application has also been submitted for the rest of the site to become a temporary car park with about 500 spaces, Fox Lloyd Jones said.
The Wellington Street building, which was opened by the Prince of Wales in 1970, once housed more than 1,300 staff.
Demolition of the building is due to be completed by June.
The newspapers' staff have moved to a new development on Whitehall Road in the city.
Plans to leave a landmark concrete tower in place after the demolition of the rest of a newspaper building in Leeds have been announced.