Trump said he doesn't "think any deal should be made on this until they approve Save America", a NewsNation reporter posted on X on Sunday, quoting Trump's remarks from an interview.
The bill requiring proof of citizenship for voter registration is called the Save America Act.
It currently lacks the 60 votes needed to overcome Democratic opposition in the 100-member Senate, where Republicans hold 53 seats.
Trump also said he was prepared to deploy agents from the US Immigration and Customs Enforcement, which is part of DHS, at airports "for as long as it takes", according to NewsNation.
Trump has said he would put ICE agents to provide security in airports until Democratic lawmakers agree to fund DHS.
Transportation Security Administration (TSA) personnel are set to miss a second full paycheck on Friday amid the partial government shutdown as lawmakers clash over funding for the DHS.
Many TSA officers have been calling in sick as paychecks have dried up, with the shortage of security agents disrupting travel at major airports.
"I don't think any deal should be made on this until they approve SAVE America," Trump was quoted as saying.
Proponents of the bill have argued it would help deter voter fraud.
Republicans have echoed Trump's false claim that large numbers of people who are in the country illegally vote in US elections.
Democrats and other critics of the bill have argued that it could disenfranchise Americans who lack ready access to passports, birth certificates and other forms of identification.
