"Democrats have been clear and consistent in our position. We are ready to work toward a bipartisan spending agreement that improves the lives of American families and addresses the Republican healthcare crisis," they wrote in a letter.
"However, we will not support a dirty spending bill that continues the Republican assault on healthcare."
There was no immediate response from the White House.
Congress, which is controlled by Republicans, failed to address the funding issue before lawmakers left Washington DC on Friday for a break.
The House approved a Republican proposal to keep the federal government funded into November but the measure failed in the Senate.
A Democratic proposal that would have boosted health care funds also failed.
It all leaves Congress and the White House with no easy way out of the stand-off that threatens a shutdown in less than two weeks when the current budget year and funding expires.
Trump's first term in office included a month-long shutdown, the longest in federal history, in 2018-2019.
Trump predicted on Friday that there could be "a closed country for a period of time".
He said the government will continue to "take care" of the military and social insurance payments in a closure.
Republicans have insisted they are not to blame for any possible shutdown, turning it back on Democrats.
House Speaker Mike Johnson and Senate Majority Leader John Thune have put forward the short-term measure, which is a typical way that Congress resolves such logjams.
That would keep government operations running at current levels as talks get underway.
While the House was able to narrowly pass the temporary funding measure on a mostly party-line vote, in the Senate the process can require a higher 60-vote threshold that means support is needed from Republicans and Democrats.
Democrats say they are working to protect health care programs.
The Democratic proposal would extend enhanced health insurance subsidies set to expire at the end of the year, plus reverse Medicaid cuts that were included in Republicans' big tax break and spending cut bill enacted earlier this year.
Republicans have said the Democrats' demands to reverse the Medicaid changes are a nonstarter but they have also said there is time to address the health insurance subsidy issue in the months ahead.
with AP
