In a post on social media, Trump said congressional Republicans shouldn't make "any deal with the Crazy, Country Destroying, Radical Left Democrats" until they agree to pass the voter bill known as the 'Save America Act'.
Trump pointed to Democratic lawmakers' demands related to a dispute over DHS funding that has left the agency unfunded since February 13.
The lack of funds has meant tens of thousands of Transportation Security Administration personnel have worked without pay for five weeks, leading some airport security workers to call in sick or quit entirely. TSA absences this weekend reached their highest level since the partial government shutdown began.
Passing the voter bill, Trump wrote, is "far more important than anything else" on the Senate's agenda, including DHS funding.
The president had already threatened on March 8 to withhold his signature from any other legislation until Congress passes the Republican-backed voting bill.
The bill currently lacks the 60 votes needed to overcome Democratic opposition in the 100-member Senate, where Republicans hold 53 seats.
Trump on Sunday also pushed for Democratic approval for other items he wants added to the bill, including banning transgender women from women's sports, outlawing "transgender mutilation of our children" and restricting mail-in ballots except in cases of illness, disability, military service or travel.
Republicans have resisted Democratic lawmakers' demands for reforms to Immigration and Customs Enforcement as a condition of DHS funding.
In an attempt to ramp up pressure on Democrats, Trump this weekend said he would put ICE agents in airports until Democratic lawmakers agree to a DHS budget bill.
Trump on Sunday said he would deploy ICE personnel at airports "for as long as it takes", according to NewsNation.
Proponents of the Save America Act have argued that it would help deter voter fraud.
Republicans have echoed Trump's false claims 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.
