Councillor Adam Davies, who represents Brierley Hill on Dudley Council, said a number of people had contacted him about the weeds at the landmark.
But, after meeting with the team at the museum, he said there wasn't a "quick fix" as the structure is still set to undergone repair work.
Councillor Davies, who posted the update on social media, said: "A number of residents have contacted me with concerns about weeds growing out the side of the Cone itself.
"Unfortunately, there isn't a quick fix to this as the Cone is still to undergo significant repair works and heritage experts have advised that works to remove the weeds before these repairs are done could further damage the Cone's structure.
"That said, I have asked if the weeds can at least be cut back in the meantime to minimise how noticeable they are. Whether this temporary cutting back is possible will depend on the advice of the heritage professionals.
"In general, it was reassuring to meet with the team and receive an update on overall plans for repairing the Cone and securing its future. It is Grade II-listed and a Scheduled Monument, so these works will take time – but, crucially, the right processes are underway.
"Although the main area is closed off to visitors, the visitor centre, shop, upstairs gallery, indoor exhibition spaces, studios and Red Cone Coffee House are still open... so make sure you pop along and enjoy more than a drive passed this local gem."