The Britain's Got Talent judge is said to be "devastated" after the split, which comes just two years after the couple welcomed their son Alfred in May 2013.
A source told the paper, "David has been putting on a brave face. But people close to him are very worried because they know he's devastated."
The couple were last seen in public together in December, when they attended the wedding of Elton John and David Furnish, and then, a few days later, watched the London production of Edward Scissorhands together.
David and Lara began dating in September 2009, and married just a few months later in May 2010 at London's Claridge's Hotel.
Guests at the nuptials included Russell Brand, Sir Elton John and Stephen Fry, while Lara wowed in a £30,000 gown.
Speaking to the Daily Mail, a source said David is now determined to win her back, saying, "David hopes it is a blip and that they can get the marriage back on track somehow, especially seeing as there is a child involved."
We hope so, too.