The 57-year-old currently works as the BBC News religion editor.
A spokeswoman for the BBC said: “We are sorry to say that Martin is seriously unwell with Covid-19 related complications.
“Everyone at the BBC is wishing him a full recovery.
“We’d ask that his privacy, and that of his family, is respected at this time.”
Bashir began working as a journalist in 1986 but made headlines around the world in 1995 for his BBC interview with Diana, Princess of Wales for Panorama.
That interview has seen renewed interest in the lead up to a new Channel 4 film, which examines the circumstances behind their meeting, airing on Wednesday night.
Bashir’s other high-profile interviews have included the suspects in the Stephen Lawrence murder case, entertainer Michael Barrymore, Jeffrey Archer and Major Charles Ingram, dubbed “the coughing major”.