The couple, who underwent a civil partnership in 2005 on the day the ceremonies were introduced, said family and friends had been eager to hear about their plans.
In an interview for US television, John 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 marriage became possible in England and Wales for the first time at the weekend. David Cameron hailed the introduction as a “historic day”.
John said: “For this legislation to come through is joyous and we should celebrate it. We shouldn’t just say, ‘Oh, well, we have a civil partnership, we’re not going to bother to get married’. We will get married.”
During the interview for NBC’s Today programme, Furnish said of the anticipation from their friends: “The phone’s been ringing off the hook.”
In a separate interview with the 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 said John had had the names of their children – Zachary, three, and Elijah, one – tattooed on his back.
Sir Elton John is back in the album charts with a 40th anniversary edition of Goodbye Yellow Brick Road, which went in at no 12 on Sunday.
More than 2,000 people and a host of famous faces gathered on Saturday to watch comedian Sandi Toksvig and her partner, Debbie, renew their civil partnership vows at a public ceremony at the Royal Festival Hall on the South Bank in London.
A number of couples vied to become the first to marry the moment the law changed.
The human rights campaigner Peter Tatchell acted as chief witness for the wedding just after midnight on Saturday of Peter McGraith and David Cabreza, who married at Islington town hall in north London after 17 years together. Tatchell said the couple, as well as the many others, had made history, leaving Britain “a more tolerant, equal place”.