The duo had been asked to perform in a new dance group on the BBC One show alongside Ian Waite, Darren Bennett and Brian Fortuna. However, both dancers have decided to pursue other opportunities instead.
Fortuna pulled out of the reality show last month.
The new dance group was introduced following the reduction in the number of celebrities taking part in this year's programme from 16 to 14, and the arrival of three new male dancers - Artem Chigvintsev, Jared Murillo and Robin Windsor.
Waite and Bennett have agreed to take part in the new dance troupe and will be joined by four new recruits, three female and one male.
Speaking about his exit, Cutler said: "I've absolutely loved my time on Strictly, and now is a good time to focus on developing new areas of my career. I'm excited about having more time to commit to new projects, and of course, still having a hand in Strictly without all the training!"
Kopylova, who has been on the show for six years, said: "I will miss it, but I am looking forward to new opportunities ahead. Of course I will be there to support [my husband] Darren and all of my fellow professionals."
Cutler joined the Strictly family in the third series and partnered celebrities including Siobhan Hayes, Carol Smillie, Christine Bleakley and Alesha Dixon, with whom he won the 2007 series title.
Kopylova's partners have included Don Warrington, Matt Dawson, Aled Jones and 2005 champion Darren Gough.
Both Cutler and Kopylova have not ruled out guest appearances on the show and are expected to be involved on an ad hoc basis in training.
Strictly returns this autumn to BBC One.