The babies are due in June, according to CBS's The Talk host Julie Chen.
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.