Trump suggested that his involvement in the design of the controversial wall would help to bring down the estimated $21 billion (£16.8 billion) it will take to construct the barrier.
"I am reading that the great border wall will cost more than the government originally thought, but I have not gotten involved in the design or negotiations yet," he tweeted on Saturday.
"When I do, just like with the F-35 FighterJet or the Air Force One Program, price will come WAY DOWN!"
Trump's original figure for the 2,000-mile-long wall was $12 billion (£9.6 billion), but a leaked report from the Department of Homeland Security estimates that the cost could be as high as $21 billion (£16.8 billion), with construction taking up to three years.
The 650 miles of fencing already in place along the border came at a price of $7 billion (£5.6 billion), and Trump's plans involve extending the barrier through mountainous terrain, which would be more expensive.
The president has repeatedly called for Mexico to foot the bill for the wall, but the country is refusing. A 20 per cent tax on all imports from Mexico has been proposed as a way to fund the project.
Earlier this month, a spoof of an Ikea flat-pack furniture kit was put forward as a tongue-in-cheek option to make the barrier affordable.
Since Trump announced the plans for the border wall during his presidential campaign, architects and designers have responded in a variety of ways.
Mexican firm Estudio 3.14 visualised the "gorgeous perversity" of the idea as a giant pink barrier influenced by the architecture of Luis Barragán, while Miami firm DOMO Architecture + Design created renderings that show a series of landscaped features along the border instead of a fence or wall.