The tabloid was launched on February 26 as a replacement for the News of the World, which was axed by News International in July following the exposure of phone hacking activities at the paper.
Taking to Twitter this evening, Murdoch predicted that initial interest in the title would wear off somewhat for its second edition.
"What will second Sunday edition of Sun sell?" he asked. "My guess down fifteen per cent would be a great result."
Murdoch claimed this week that The Sun on Sunday had sold 3m copies upon its debut.
However, many media commentators felt underwhelmed by the publication, with Matthew Engel of The Sunday Times calling it "timid, mumbling, joyless and dull".
Roy Gleenslade, meanwhile, described The Sun on Sunday as "bland", saying: "What struck me is the paper's soft focus. Everyone is mentioned in a very kind way. There is a clear intention to draw a line under the News of the World and that type of intrusive journalism."
The Sun on Sunday counts Katie Price, Heston Blumenthal, Nancy Dell'Olio and political writer Toby Young among its regular columnists.
It was announced on Wednesday (February 29) that James Murdoch, son of the News Corporation CEO, is to step down as executive chairman of Sun publisher News International in the UK.