President Barack Obama set the refugee cap at 110,000 in 2016, his final year in office.

Resettlement agencies have until Jan. 21 to submit to the administration their funding proposals, with the consenting letters from state and local governments.

Mr. Trump has made the restriction of both legal and illegal immigration a central mission of his first term. In addition to cutting the number of refugees allowed into the country, he has also restricted the ability of those who approach the southwestern border to obtain asylum. The administration is also considering adding additional countries to Mr. Trump’s travel ban, according to three people familiar with the discussions.

Mr. Abbott’s refusal of refugees could have far-reaching effects. More than 16,700 refugees resettled in Texas from October 2015 to September 2019, more than any other state, according to Lutheran Immigration and Refugee Service, a resettlement agency.

Krish O’Mara Vignarajah, the chief executive of that agency, said Mr. Abbott’s decision would jeopardize the safety of families around the world.

“Nearly 2,500 refugees started to rebuild their lives in Texas last year, many of whom have additional family members in harm’s way seeking to join them in safety,” Ms. Vignarajah said in a statement. “These families have been torn apart by violence, war and persecution — but we never thought they would be needlessly separated by a U.S. state official.”

Many communities have said they would open their doors to refugees, including 42 states and at least 90 local governments, according to a spokesman for Lutheran Immigration and Refugee Service, which has been tracking the number of letters consenting to refugee resettlement. Mr. Abbott issued the fourth refugee rejection under the executive order. Leaders in Appomattox County, Va., and Beltrami County, Minn., also rejected refugees, although neither county is a destination point for refugees, according to Ms. Vignarajah.

The mayor of Springfield, Mass., which has a sizable refugee population, also said he will reject refugees. “It’s time for other much more affluent communities to take on their fair share,” Mayor Domenic Sarno told The Republican newspaper, saying that “Springfield is at capacity.”