Mr Bashir, who made global headlines in 1995 after interviewing Princess Diana for the corporation’s Panorama programme, is currently the BBC News religion editor.
A BBC spokesperson 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."
Nine years after he started working as a journalist, Mr Bashir became well-known for a 1995 interview in which Diana admitted to having had an affair with her riding teacher, James Hewitt.
The Princess of Wales also spoke about Prince Charles’s infidelity. "There were three of us in this marriage, so it was a bit crowded," she said, referring to Camilla Parker-Bowles, who later married Charles in 2005.
The interview, which was watched by 15 million people, has been put into the public spotlight again this week, after a Channel 4 programme about it aired on Wednesday evening.
The 57-year-old journalist has also conducted high-profile interviews with other celebrities including several with Michael Jackson in 2003.
Before returning to the BBC, he worked as an anchor at ABC and then MSNBC, resigning from the latter position in 2013 after making derogatory comments about the former vice-Presidential candidate Sarah Palin.
In October last year, Mr Bashir competed in a celebrity version of the X Factor, later telling ITV’s Good Morning Britain programme that some of his colleagues had sneered at him for going on the show.
“It has been humiliating. Some of my colleagues at the BBC have sneered a bit and said, ‘Oh you look like an idiot, what an embarrassment,’" he said.