The couple, who underwent a civil partnership in 2005 on the day such ceremonies were introduced, said friends had been eager to hear about their plans.
In an interview for US TV, Sir Elton said: "We'll do it very quietly. But we will do it and it will be a joyous occasion and we will have our children."
Gay marriages became possible in England and Wales for the first time at the weekend. Prime Minister David Cameron hailed the introduction as a "historic day".
Sir Elton - who has two children with Furnish - said: "For this legislation to come through is joyous, and we should celebrate it."
And during the interview for NBC's Today pro-gramme, Furnish said of the anticipation from their circle of friends: "The phone's been ringing off the hook."
In another interview with Las Vegas Review-Journal, Furnish said: "I think what we'll do is go to a register office in England in May, and take the boys with us, and a couple of witnesses."
He also told how his chart-topping partner has had the names of their children tattooed on his back.
Sir Elton is back in the album charts with a 40th anniversary edition of Goodbye Yellow Brick Road.