The newspaper reported that the 31-year-old model has left home with the couple's two-year-old son Alfred along with their pet dog.
Britain's Got Talent judge and children's book author David, 43, was said to be 'devastated' at the split. The Sun says the comedian has agreed to a trial separation.
It quotes a source saying: "David has been putting on a brave face. But people close to him are very worried because they know he's devastated."
Model Lara married the Little Britain star in 2010 but it is thought the pair have drifted apart. The last time they were spotted together was at Sir Elton John's wedding to David Furnish on December 21.
Two days later they met the London theatre cast of Edward Scissorhands, where the last public photograph of the couple together was taken.
Lara's dad has told how besotted David set out to woo the model by romantically filling her home with flowers leading to their first date at a Chelsea football match in March 2009.
They announced their engagement on January 20, 2010 and wed at Claridge's hotel that May.
Two years later Lara was pregnant and Alfred arrived on May 6 2013.
When she was pregnant, she posed for a stark naked full-length painting for artist Marc Quinn – lying on mounds of red meat.