Mr Trump is also asking to speed up deportations and hire thousands of new immigration officials.
Last month he ended the Deferred Action for Childhood Arrivals (Daca), that protected nearly 800,000 immigrants.
Leading Democrats in Congress immediately rejected the new plan.
The list of "principles" - delivered by the White House to Congress on Sunday - includes the following requests:
- Construction of the wall across the border with Mexico
- Employing 10,000 additional Immigrations and Customs Enforcement officers and 1,000 attorneys for the agency
- An extra 370 immigration judges and 300 federal prosecutors
- Banning immigrants from bringing their extended family members to the US
- The use of the E-Verify programme by companies to keep illegal immigrants from getting jobs
"These priorities are essential to mitigate the legal and economic consequences of any grant of status to Daca recipients," Mr Trump's legislative affairs director Marc Short said.
The Daca programme was a scheme set up in 2012 during the presidency of Barack Obama to protect young undocumented immigrants who were brought to the US illegally as children - so-called Dreamers (mostly Latin Americans) - from deportation.
They were able to apply for work and study permits under a policy which, critics said, amounted to an amnesty for illegal immigrants.
In September, Mr Trump told Congress it had six months to agree new legislation to help the Dreamers.
His new proposals were swiftly rejected by leading Democrats.
"The administration can't be serious about compromise or helping the Dreamers if they begin with a list that is anathema to the Dreamers, to the immigrant community and to the vast majority of Americans," House of Representatives Democratic leader Nancy Pelosi and Senate Democratic leader Chuck Schumer said in a joint statement.
"The list includes the wall, which was explicitly ruled out of the negotiations. If the president was serious about protecting the Dreamers, his staff has not made a good faith effort to do so," they said.
Many Daca recipients protested against Mr Trump's decision to scrap the programme
What is Daca?
The Daca programme protected roughly 750,000 people in the US from deportation and provided temporary permits for work and study.
In order to qualify for Daca, applicants under the age of 30 submitted personal information to the Department of Homeland Security.
They had to go through an FBI background check and have a clean criminal background, and either be in education, recently graduated or have been honourably discharged from the military.
In exchange, the US government agreed to "defer" any action on their immigration status for a period of two years.
The majority of so-called Dreamer immigrants in the US are from Mexico and other Latin American countries.