The pair became buddies after meeting at one of Gaga's shows in London in May, and are said to have discussed their mutual love for the instrument.
And although Gaga, 25, won't be able to make it to the supermodel's wedding to Jamie Hince next month, Kate is said to be thrilled with the quirky gift.
A source told the Daily Star, 'Kate was so touched that Gaga sent her such a thoughtful present.
'Kate would have loved Gaga to be at the wedding, but because of her busy work schedule she can't make it.
'But Gaga is hoping to stay with Kate at her Cotswolds home later this year.'