South Dakota Governor Kristi Noem wrote to several tribal leaders last week saying the checkpoints were illegal.
But the Sioux say they are the only way of making sure the virus does not enter their reservations.
Their limited healthcare facilities would not be able to cope with an outbreak, they say.
At present, people are only allowed to enter the reservations for essential business if they have not travelled from a Covid-19 hotspot.
They must also complete a health questionnaire before doing so.
Ms Noem is threatening to take the two tribes - the Oglala Sioux and the Cheyenne River Sioux tribes - to federal court if they do not comply.
In a letter sent to their representatives on Friday, she demanded the checkpoints be removed.
"The checkpoints on state and US highways are not legal, and if they don't come down, the state will take the matter to federal court, as Governor Noem noted in her Friday letter," her senior adviser and policy director, Maggie Seidel, said in an email sent to the local Argus Leader newspaper on Sunday.
Tribes are meant to get permission from state authorities if they want to close or restrict travel inside their reservations.
The chairman of the Cheyenne River Sioux tribe, Harold Frazier, issued a statement in response to the governor on Friday, saying: "We will not apologise for being an island of safety in a sea of uncertainty and death."
"You continuing to interfere in our efforts to do what science and facts dictate seriously undermine our ability to protect everyone on the reservation," he added.
Oglala Sioux President Julian Bear Runner says Ms Noem's decision "threatened the sovereign interest of the Oglala people".
"Due to the lack of judgment in planning of preventative measures in response to the current pandemic, Covid-19, the Oglala Sioux Tribe has adopted reasonable and necessary measures to protect the health and safety of our tribal members and our other residents of the Pine Ridge Indian Reservation," he is quoted as saying by the Argus Leader.
Mr Frazier says the main purpose of the checkpoints is to monitor and try to track the virus. "We want to ensure that people coming from 'hotspots' or highly infected areas, we ask them to go around our land," he told CNN.
"With the lack of resources we have medically, this is our best tool we have right now to try to prevent [the spread of Covid-19]," he added.
He says the reservations are ill-equipped to deal with a coronavirus outbreak, with the nearest critical care facilities three hours away.
The Cheyenne River Sioux Tribe only operates an eight-bed facility on the reservation and no intensive care unit for the 12,000 people that live on the reservation, he adds.
No stay-at-home order
South Dakota is one of a handful of US states which have not issued stay-at-home order to their residents.
There were 198 cases of Covid-19 among Native Americans in the state as of Sunday, according to state health department figures quoted by CNN. The state has more than 3,500 confirmed cases and at least 34 deaths, according to Johns Hopkins University.
The US has the highest number of virus deaths and cases in the world - but it also has one of the biggest populations, and widespread testing.