Rothko, who died in 1970, is renowned for his large abstract paintings featuring bold blocks of color.
The defaced painting was one of a series intended to decorate the Four Seasons restaurant in New York. Rothko changed his mind about the commission and gave the works to galleries, including the Tate.
The artist's children, Kate Rothko Prizel and Christopher Rothko, said in a statement that they were "greatly troubled" by the incident, but confident the Tate would do everything it could to remedy the situation.
While the Tate Modern has said it does not have a price for the defaced piece, another Rothko piece — "Orange, Red, Yellow" — sold for almost $87 million at auction in New York.
This is not the first time an artwork at Tate Modern has been interfered with. In 2000, two Chinese performance artists attempted to urinate on Marcel Duchamp's urinal sculpture "Fountain."
Jill Lawless and Raphael Satter in London contributed to this report.