"The wall will be built where it's needed first, and then it will be filled in".
"I am pleased to be working with Secretary Kelly on solving our border challenges and am encouraged by the attention the federal government is giving this serious issue", said Governor Abbott.
"I really hope to have it done within the next two years", the retired four-star Marine general added.
There will be a multi-layered approach to building the wall - with a physical wall and part of the wall "you can see through because it will rely on sensors and technology", Kelly said on Thursday.
He has said that the wall will cost around $8billion (£6.3bn), although wall-builder Todd Sternfield has suggested that it is more likely to come in at $26billion (£20.6bn).
During his presidential campaign, Trump repeatedly said that he will build a wall along the US-Mexico border to control immigration.
But the idea of the wall being more of a transparent barrier between the two countries, at least in part, contradicts Trump's claim that he will erect a "big, handsome wall" along the entire length of the border.
Aside from touring different sites, the officials also took the opportunity to personally meet with men and women who work on the front lines of border security.
The DHS head talked about another recent controversy too - the rocky rollout of the travel ban from seven Muslim-majority countries. "I think that alone would act as a huge deterrent for people who are considering making the trip up", he said.
Both Kelly and Trump have said they believe they have the authority to begin work under existing law. In addition to Ryan's import tax, those options could include an increase in visa and border-crossing card fees, or a lump-sum payment from Mexico of billions of dollars, according to Fox News.