In a letter to the State Department on Friday, Abbott said that Texas had done "more than its share in assisting the refugee resettlement process", and argued that the state should be focused on "those who are already here, including refugees, migrants, and the homeless", the BBC report said.
On that basis, Abbott said he "cannot consent to initial refugee resettlement" in 2020, but added that the decision "does not deny any refugee access" to the US.
Refugees who are already settled in other states, Abbott said, would be allowed to move to Texas if they choose.
Texas has large refugee populations in several of its major cities. In the 2018 fiscal year, Texas took in 1,697 refugees - more than any other state, but a large drop from 4,768 in the previous fiscal year.
Abbott's decision means Texas will become the first US state known to do so, the BBC report added.
In September 2019, President Trump announced that states must actively consent to any resettlement of refugees after June 2020.
So far, the Governors of more than 40 other states have said they would opt in to the government programme.
Trump has made reducing immigration a key aim of his administration. He has slashed the number of refugees allowed into the country for the 2020 fiscal year to 18,000 - a record low.
About 30,000 refugees were resettled in the US during the previous fiscal year.
The previous lowest admissions figure was in 2002, after the 9/11 attacks, when about 27,000 refugees were allowed into the US.
( With inputs from IANS )