According to The Sun, Corden is "not really her type", but she is currently in text contact with the comedian.
"He is absolutely besotted. It's a bit full on but Alexandra seems to be taking it in her stride. I wish he'd give me a Cartier watch too!" said Burke's mum Melissa Bell.
A source added: "Alexandra is massively flattered and overwhelmed. A gift like that is hard to ignore."
The Gavin & Stacey actor has been romantically linked to a string of women in 2008 including Sheridan Smith, Lily Allen, Leona Lewis and Alesha Dixon.