The Beatles legend, who was spotted enjoying a romantic walk in London with Rosanna, invited the actress to his East Sussex mansion to meet four-year-old Beatrice.
A friend of Paul's told Britain's The Sun newspaper: "For Rosanna to be welcomed into Paul's inner sanctuary means he thinks a great deal of her.
"They are an item. They have been together since August. Why else would he take her to his country estate to meet his little girl? He is checking her out."
The 65-year-old musician is said to be so smitten with the 'Desperately Seeking Susan' star he has asked his female friends for advice on how to make the romance last.
Heather Mills, who is currently embroiled in a bitter divorce battle with Paul, is said to be furious about Rosanna, 48, meeting the couple's daughter.