"We realise that there is a lot of hard work to be done and we have to be prepared to do that."
Beckham, who admitted to being "star-struck" after meeting with Uefa president Michel Platini, added: "It would mean so much to get it.
"To have the World Cup in our country, to take my sons to the World Cup would be special and I think there are many fathers that will be thinking that."
Italian Capello also threw his weight behind the 2018 campaign, saying: "I think England is the best place to have the (2018) World Cup.
"It has fantastic stadiums and people who love football - it's the home of football."
England are one of nine bids for either the 2018 or 2022 tournaments, along with Russia, Australia, the USA, Japan, Mexico, Indonesia and joint bids from Portugal-Spain and Netherlands-Belgium. Qatar and South Korea are also making bids solely for the 2022 staging.
But there was controversy when the arrival of Luis Figo, ambassador for the Portugal-Spain campaign, interrupted the USA's video presentation of their bid.
It led to a rush of reporters and photographers leaving the USA's video right in the middle of the presentation.
"It was totally ruined," Major League Soccer commissioner Don Garber said. "I think there's an element of fair-play in our sport. That was bad form."
England should win 2018 bid on merit - Dein
The Portugal-Spain bid later apologised with a spokesman saying: "We are very sorry - this was not intended."
According to former Arsenal vice-chairman David Dein, England's bid remains strong despite a recent series of "own goals".
"There's a year to go," Dein told BBC Sport. "We still have a great chance."
While Beckham has brought plenty of glitz to England's bid, Dein could yet to play a key role in helping to deliver vital Fifa executive committee votes, despite not having an official role at present.
Last month Dein, who has a network of worldwide contacts, travelled to Nigeria for the Under-17 World Cup - a competition where six Fifa executive members were in attendance - following an invitation from Fifa and Confederation of African Football member Amos Adamu.
And Dein, who has been attending the Soccerex conference in South Africa, signalled that he would be willing to take on a more active role if asked by the Football Association.
"Talks are ongoing and I'm ready to help my country if called upon to do so," said Dein, who was a director of Arsenal for more than 20 years.
There are still a lot of goals to score against the opposition
Former Arsenal vice-chairman David Dein
England's 2018 bid has suffered with its public relations and internal feuding at board level, hence Dein's reference to "own goals."
The 2018 bid team has been criticised for purchasing luxury handbags, one for the wife or partner of each of the 24 Fifa voters, with Fifa vice president Jack Warner returning the gift, describing his wife as "sacrosanct".
And Premier League chairman Sir Dave Richards resigned from the board just weeks after six other board members stood down, while Karen Brady, one of those to quit her post, has stated the bid could be derailed by "infighting".
"What's going on is heavy politics and that needs to be resolved," said Dein, who in the past has been on the Football Association's main board as well its international committee.
"It's not an easy situation. There have been a lot of changes and sadly a few own goals scored.
"It's important we start rowing to the same beat, but there are still a lot of goals to score against the opposition.
Fifa's 24-member ruling executive will choose the hosts next December.
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