Demonstrations over what activists at the University of Cape Town call a symbol of white oppression have gathered strength over the past week since a student threw a bucket of human excrement over the statue.
The university campus was built on land donated by Rhodes, a mining magnate and champion of British imperialism in the late 1800s, who gave his name to the former Rhodesia, now Zimbabwe.
This week swastikas and pictures of Hitler appeared on campus buildings, drawing parallels between Rhodes and Nazi Germany's racial policies, but they were removed after protests by Jewish groups.
Price has agreed that the statue should be moved from its prominent position on campus, noting "the many injustices of colonial conquest enacted under Rhodes' watch".
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But, he said, the decision had to be taken by the university council, and he proposed talks involving students and staff ahead of a council meeting on April 15.
The students have demanded that he name a date for the removal of the bronze statue, which they say symbolises the lack of racial transformation in South Africa since the end of apartheid 21 years ago.
"So we want it to happen now, not in five years, not another talk. Now is the time."
Students have dismissed Rhodes' campus bequest as a justification of the statue, saying the land was stolen from black Africans in the first place.
Monuments to South Africa's racist white-minority rule are scattered throughout the country and are regular targets of protest.
But the discontent among some black university students goes beyond symbols to cover admission policies and the racial make-up of the teaching staff.