Scotland Yard has launched an investigation after the mural, one of Rothko's Seagram series, was defaced Sunday with what appears to be the words "Vladimir" and "a potential piece of yellowism."
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.
Vladimir Umanets, who identifies himself as the co-founder an artistic movement he calls "Yellowism" says he is behind the graffiti. According to an online manifesto, Yellowism is an artistic movement run by Umanets and another person, Marcin Lodyga.
Umanets told Britain's Press Association news agency on Monday that he wants to draw people's attention to his movement, which he describes as "an element of contemporary visual culture."
"The main difference between 'Yellowism' and art is that in art you have got freedom of interpretation, in 'Yellowism' you don't have freedom of interpretation, everything is about 'Yellowism,' that's it," he said.
Umanets told Press Association he expects to be arrested, but believes his scrawl increased the painting's value.
"I believe what I am doing and I want people to start talking about this. It was like a platform," he said. "I didn't decrease the value, I didn't destroy this picture, I put something new."
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 in London contributed to this report.