Trump's comprehensive plan - which will be formally presented next week - includes eliminating the popular "green card lottery" program and would severely restrict family immigration.
It would also require Congress to budget $25 billion for a "trust fund" for constructing a wall on the US-Mexico border - a major plank of Trump's White House campaign.
"The Department of Homeland Security must have the tools to deter illegal immigrants; the ability to remove individuals who illegally enter the United States, and the vital authorities necessary to protect national security," a senior White House official told journalists.
The White House's offer of a path to citizenship for the Dreamers was much wider than expected. Earlier it had suggested it was only open to granting citizenship to the 690,000 young immigrants registered under the Deferred Action for Childhood Arrivals, or DACA, program.
But in exchange, Trump was asking Congress to make future legal immigration more difficult - and to shore up the Homeland Security Department's toolbox and funding to crack down on the overall population of unauthorised immigrants, estimated at some 11 million including Dreamers.
That proposal immediately provoked the ire of Democrats, who have been battling Republicans and Trump for weeks over immigration.
"$25 billion as ransom for Dreamers with cuts to legal immigration and increases to deportations doesn't pass the laugh test," said Democratic Representative Luis Gutierrez.
"Dreamers are being held hostage by Republicans who want to tie any solution to their extremist immigration policies," said fellow congressman Jerrold Nadler.
