Supporters of the plan say it will be key to tackling illegal immigration, likely to be a key issue in next month's mid-term elections.
But Mexican officials have opposed the fence, with outgoing President Vicente Fox calling it "shameful" and likening it to the Berlin Wall.
About 10m Mexicans are thought to live in the US, some 4m of them illegally.
An estimated 1.2m illegal immigrants were arrested last year trying to cross into the US via the border states of Texas, New Mexico, Arizona and California.
The BBC's Nick Miles in Washington says that, though few US congressmen have questioned the need for some action to reduce illegal migration, many have queried how effective the fence will be.
TJ Bonner, president of the National Border Patrol Council, a union representing patrol agents, told Associated Press that it would not be enough on its own.
"A fence will slow people down by a minute or two, but if you don't have the agents to stop them it does no good. We're not talking about some impenetrable barrier," he said.
Mr Bush himself believes that tougher border security should be accompanied by moves to help legalise the status of the illegal migrants currently in the US.
But those moves are opposed by many within his own Republican party.
Mexico has promised to challenge the fence at the United Nations and on Wednesday presented a declaration against the policy to the UN's Human Rights Council, supported by 27 other Latin American and Caribbean nations.
Part of the funding for the fence is likely to come from the $1.2bn (?0.6bn) set aside for it in a recent homeland security bill, but the full cost may be greater and the source of the funding is still unclear, according to our correspondent.
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What on earth are they thinking? They are throwing money away here! It's disgusting. This is in no way going to solve illegal immigrants coming through. It's only going to make it look like they're trying to do something about it!
After having to deal with my husband going through the immigration process I can tell you the biggest problems are internal. You call to ask a question and are told one thing. You call back the next day and ask the same question because you are still confused and explain the answer you were given and are told that law is no longer there but it's something else!
The right hand doesn't know what the left is doing!One time when we were crossing through Border Patrol in El Paso my husband gave them his green card and they insisted he was registered as the wrong type of immigrant! We argued about it for about 10 minutes before they finally let us through. When I got home I looked it up and found out the Border Patrol Officer (who is supposed to be protecting us and know what to do) was absolutely clueless!
Billion dollar fence. How many schools could that money fund?
