A Guard spokesman said 250 personnel would be sent to patrol the area within the next 72 hours.
Arizona is also planning to deploy 150 troops there next week.
President Trump says he wants to send up to 4,000 National Guard members to secure the border with Mexico, until his proposed border wall is built.
The states of New Mexico and California have been asked to take similar action to Texas and Arizona.
Also on Friday, the US president outlined plans to end a practice dubbed "catch and release" as part of his stricter anti-immigration policies.
Mr Trump wants illegal immigrants to be held in detention while they wait to hear if they will be deported, instead of being freed.
He has asked the US Department of Defense for a detailed list of military and other facilities that could perform that function.
Mr Trump has sent several tweets over the past seven days railing against illegal immigration, and accusing Democrats of allowing "open borders, drugs and crime".
He declared on Twitter that Republicans should "go to Nuclear Option to pass tough laws [on illegal migrants] NOW".
Mr Trump also threatened Mexico, saying the North American Free Trade Agreement (Nafta) was at risk unless it stopped the movement of migrants over the border.
Mexico's President Enrique Pena Nieto has condemned what he called "threatening or disrespectful attitudes" from Mr Trump.
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