The two stars have grown increasingly close since the Twilight actor split from Kristen Stewart for a second time three weeks ago, and after initially offering the British star support in the wake of his break-up romance is blossoming between the pair.
Katy - who recently ended her romance with John Mayer - and Rob went to see the Icelandic singer perform at the Pantages TheatrE on Saturday but left separately to avoid paparazzi and the date follows the British star spending a week at Katy's house.
A source told The Sun newspaper: "They are both good friends but have been hanging out more and more recently.
"Rob poured out his heart to Katy about Kristen cheating on him. She proved to be a really good shoulder to cry on.
"He stayed at her house in LA after they went away to Santa Barbara together for few days over American Memorial Day weekend late last month. Their friends say they have now fallen for each other."
Rob, 27, and Katy, 28, are now believed to be planning a Caribbean vacation together as their relationship blossoms.
The source added: "They are talking about going to the Bahamas in the next few weeks to visit some friends of Katy's. They are both hopeless romantics, so it wouldn't be a surprise if things developed quickly."
Rob dumped Kristen - who he had been in a four-year relationship with - for a second time after discovering she was back in touch with Snow White and The Huntsman director Rupert Sanders, who she had an affair with last year.
Katy was previously married to British comedian Russell Brand but they union ended in December 2011 after 14 months and their divorce was finalised last July.
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