The reality TV contestant, 30, reportedly angered his new boss by calling in sick yesterday.
"This will make Sir Alan really cross because he hates being let down," a source told The Sun.
"Lee was really keen to start work but he felt he wasn't up to it and had to stay at home."
McQueen beat 15 contestants to a £100,000-a-year job helping develop digital display advertising last month.
A spokesperson for Sir Alan Sugar's holding company Amshold said: "Lee is at home with a virus. He'll be at work as soon as he is well."