Robert recently took in a rescue dog called Bear and although the couple - who play lovers Edward Cullen and Bella Swan in the 'Twilight: Saga' film franchise - refer to the pooch as their "baby", they are keen to co-adopt another animal.
A source said: "It's really sweet. When Kristen returns to LA from the UK where she's filming 'Snow White and the Huntsman', I think they'll adopt a new pet. They want to pick one together."
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The couple are hoping to set up home together soon and although they are undecided on whether to settle in London or Los Angeles, they are hoping to purchase a holiday home on the Isle of Wight where they have spent the last two New Year's Eves.
The source added to More! magazine: "Because neither of them could cook, they did some Italian cooking lessons together earlier in the year. I think that's their idea of bliss - sitting on the sofa together at home after making a really great dinner."