Another source close to the couple, quoted by People, said they were "very happy". The Clooneys' representatives have not yet commented.
The couple married in Venice in 2014 with a star-studded list of guests who included Matt Damon and Bill Murray.
Rumours that they might be expecting began to circulate last month when Mrs Clooney was spotted wearing a loose-fitting dress.
Mr Clooney, 55, has starred in numerous Hollywood films and has won two Academy Awards.
Mrs Clooney, 39, has represented a number of high profile figures, including former Ukrainian Prime Minister Yulia Tymoshenko and Australian Wikileaks founder Julian Assange.