He's bombarded the Kiss Kiss chanteuse with flowers and text messages, and has even invited her to stay at his house in Los Angeles.
A showbusiness insider says, "There were sparks between the pair and it seemed like Robbie wanted to get to know her better."
However, Robbie may not be making much progress.
A pal of Holly's says, "She hasn't replied to his messages and doesn't seem to be responding to his advances.
"He's not used to being turned down, so he won't give up easily."
Stunning Holly has a long term boyfriend in Australia, PETER VEVERIS.