The Rocket Man star became a dad on Christmas Day when baby Zachary was born via a surrogate.
Speaking to UK newspaper Sunday Mirror, he said: 'There are no plans to add to the brood - we're getting used to having one baby. And Lady Gaga isn't godmother - that's just speculation - we haven't even decided on anything like that yet.'
Before the Oscars, where Elton hosted his annual AIDS Foundation Oscars Viewing Party, took place, the new family took a holiday in Hawaii.
He said: 'We got back Saturday morning from our first family holiday and it was fantastic. Zachary went on the beach, in the pool and on a hike with us. It was fantastic and he just loved it.'