The 'Patience' singer and the comedienne are planning to drive the along the iconic Route 66 from Chicago to Los Angeles, which spans 2,500 miles and will see them cross the states of Illinois, New Mexico, Texas, Arizona and California over the festive period.
A source told the Daily Mirror newspaper: "Jason and Catherine wanted to get away from it all now their relationship is out in the open.
"They wanted somewhere off the beaten track where they can spend some quality time together, so decided to take a three-week vacation.
"They both loved the idea of a long break with just the two of them and the road. Though they're also planning a last minute getaway to a Caribbean island."
Catherine, 43, and Jason, 41, have been dating since earlier this year, after they met when filming a spoof sketch for charity BBC Children in Need, in which she portrayed him.
The source added: "They don't want to rush anything and dashing off on holiday is a massive step."
When the couple initially got together, friends described them as well-suited to each other.
A friend said: "It's a very unlikely pairing but they have really hit it off. They've been on a number of dates to the theatre and the cinema. They really enjoy each other's company."