The You and I singer has also asked her interior design team to transform the chauffeur's spare room into a nursery for his arrival.
A source told British newspaper The Sun: 'Gaga is very generous with her presents and makes a huge effort for those who look after her. As soon as she found out her UK driver was going to be a dad, she was on the phone, trying to sort out a gift. It had to be something quirky, too - it's part of her nature.'
Gaga, 25, real name Stefani Germanotta, will be swapping Christmas presents with her boyfriend Taylor Kinney, when he flies from Pennsylvania to New York on Boxing Day to spend time with her and her parents.