Although the building is not a major moneymaker for the Trumps — their company manages the property and owns some portions of it, but there are hundreds of individual unit owners — it has long held symbolic importance to the president: It was the first hotel to bear his name (and now the only one left in New York), and even as “Trump” has disappeared from other buildings in New York and elsewhere, this signature property was considered untouchable.

That changed after a contentious meeting in June between the condo board and several dozen owners, some of whom complained that the Trump name had been a drag on the property ever since Mr. Trump announced his bid for the presidency four years ago, according to people with knowledge of the meeting who spoke on the condition of anonymity because it was a private session.

In heavily Democratic New York, Mr. Trump’s politics most likely contributed to the discord, which had surfaced in the past as well. But in June, some owners suggested that the polarizing nature of the Trump name was depressing the value of their investments in the building, said the people with knowledge of the meeting. With sales of units in the building slowing in recent years, some owners called for stripping the T-R-U-M-P letters from the property’s signage altogether and renaming the property One Central Park West.

The compromise plan from Donald Jr. would most likely remove the block-lettered “Trump International Hotel and Tower” sign above the building’s shiny entrance and replace it with signage that draws a clear distinction between the hotel and the tower, the people with knowledge of the plan said. That would involve elevating the presence of the One Central Park West address, which currently appears in much smaller lettering than the Trump name elsewhere on the property.

The two other existing Trump signs, both at street level, would also be modified to reflect the branding change, though the details of that modification are unclear.