The 'Angels' singer has reportedly told friends he "sees a lot of himself" in Lily and wants to spend an evening with her when she is in Los Angeles, where he lives.
Robbie hopes to take the 23-year-old singer - who has recently been criticised for her drunken antics - to his favorite restaurant STK before heading for a night of star gazing for spaceships.
A friend of Robbie, 34, told Britain's Daily Star newspaper: "Robbie loves Los Angeles but the one thing he does miss is the British sense of humours and impudence, so Lily's attitude appeals to him. He thought it would be nice to meet up while she's recording her album in town and got his people to call Lily's people to invite her for dinner at his favourite restaurant."
"Rob likes to be a big brother figure to celebrities who seem to find themselves in the same position he has been - young and vulnerable and in the glare of the media. And he feels that taking an interest in UFOs and the unknown is a good way of putting things in perspective, hence asking Lily to join him."
Earlier this week, it was revealed Robbie has taken up camping in the hope of seeing an alien.
A source close to the star said: "Robbie has been taking pals out to the woods for long weekends. He wants to live like a cowboy in a Western. He also has a fascination with UFOs and thinks he has more chance of seeing one away from Los Angeles' bright lights."
Earlier this year, it was claimed Robbie was planning to splash out $4.9 million on an alien observatory after spending all his free time visiting Arizona's National Optical Astronomy Observatory, in Tucson.
He has previously claimed to have been visited by extra-terrestrials on more than one occasion.
Describing one close encounter of the third kind, he said: "I was lying on my sun lounger outside at night. Above me was a square thing that passed over my head slightly and shot off."