The proposals would see a 20-metre-tall phone mast installed outside the 17th century Wollescote Hall, in Stevens Park, in Wollescote, between Stourbridge and Halesowen.
Councillor Tim Crumpton, who launched the petition, argues this would "destroy" the amenity of the area and has called for the plans to be withdrawn.
A planning application has been made on behalf of the Government - which wants the West Midlands to become the UK's first "large-scale test bed" for 5G, according to planning documents.
Councillor Crumpton, who represents Cradley and Wollescote, said the mast would be "erected slap bang in front of Wollescote Hall".
"This would completely destroy what is - in my opinion and in many others - the most beautiful sight we have when looking at the hall and its wonderful gardens," he said. "We must fight this."
The plans have been labelled with a special status, called a "permitted development", which means they are granted planning approval without the need to submit a planning application, said Councillor Crumpton.
However a spokesman for Dudley Council, which is the planning authority, said the plans have not yet been rubber stamped.
Councillor Ian Kettle, cabinet member for regeneration and enterprise, said: "We are currently assessing an application for a new 5G mast near to Wollescote Hall.
"No decision on this application will be made until it is assessed in line with both national and local planning policy and any comments from local residents are considered."
Wollescote Hall was built in the 1600s. It was owned by a number of families including the Millwards, for 340 years, and most recently the Stevens.
In 1930, the hall was given to the people of Lye and Wollescote by Ernest Stevens, who was a wealthy industrialist.
According to historians, the name Wollescote derives from an Anglo-Saxon name. Records show the area was inhabited as early as 1282.
Councillor Crumpton says he would be open to discussing other potential sites in Wollescote for the 5G mast.
The new technology will allow for a much faster mobile network for residents in the region.
But the masts have not come without controversy. Alongside causing eyesore concerns, the masts have also been attacked over fake 5G coronavirus claims.
In late April, the Express & Star reported 25 masts had been attacked around the West Midlands.
To sign the petition online, visit the council's website here.