"We object to proposals to install such a monument anywhere," the foundation said.
The extravagant sculpture designed by artist Dmitry Kavarga won a contest for a monument to Yeltsin held by ART4.RU, a museum of contemporary art in Moscow. The museum's director called the planned work "a black biomorphic monster symbolizing destruction and chaos consuming order."
The family of Russia's first democratically elected leader said they "did not take any part in selecting the sculpture and the winner."
The museum said Kavarga had been selected via an open popular vote involving 6,000 people.
Explaining the choice, museum director Igor Markin said the Yeltsin era went down in history as a time of destruction.
Yeltsin, praised by many for pioneering democratic change, but also widely criticized for impoverishing millions through hasty, corruption-ridden reforms in the 1990s, died on April 23 this year of heart failure at the age of 76.
Markin said the sculpture would be built on a very large scale, and be made of black metal. He claimed it would make an impression comparable to that of the Eiffel Tower.
The contest organizers plan to seek permission to build the giant work of art on Moscow's Lubyanka square in front of the former KGB headquarters.
Yeltsin is also likely to leave his mark in Yekaterinburg, where he was born and raised. The legislature and the city hall have been considering naming one of the city's streets, parks, or squares after the late leader, who already has a memorial board on the house where he lived.