The 58-year-old, who has been a familiar figure on TV news bulletins for many years, is said to be optimistic about a positive outcome.
The Sri Lanka-born journalist has been the face of BBC1's News At Six since 2007.
In a statement the BBC said: "He is grateful for all the good wishes he has received thus far and is optimistic for a positive outcome. George asks that he and his family are given the space and privacy they require whilst he recovers.
"Our thoughts are with him and his family and we send them our very best wishes during this time.
"Until such time as George is well enough to return to work the BBC News at Six and GMT on BBC World News will be presented as usual by familiar faces from BBC News."
Alagiah first began hosting the 6pm news bulletin in early 2003, but he stepped up to front it solo four years later following the departure of his co-host Natasha Kaplinsky.
He has previously been a prominent foreign correspondent, often as a specialist in Africa with coverage of civil wars in Somalia and Liberia, as well as the genocide in Rwanda 20 years ago.
Alagiah joined the BBC 25 years ago after working as a print journalist and he has gone on to write a number of books including A Home From Home, which looked at what it means to be British.
He has also presented other shows such as Mixed Britannia, looking at the UK's mixed-race population.
He was appointed Officer of the Order of the British Empire (OBE) in the 2008 New Year Honours.
Bowel cancer is the third most common type of cancer in men, behind prostate and lung, with almost three-quarters of cases affecting people over the age of 65.
If caught in its earliest stages, the chances of surviving for a further five years are 90 per cent, and it is possible that patients can be given the all-clear.