Trump is demanding funding for a wall on the Mexican border.
"Democrats will be offering the Senate Republican appropriations legislation to re-open government later today", Pelosi said.
The new congress is considered the most diverse in history with record numbers of women and women of colour.
Pelosi, a Democrat, sent the letter to the Republican president hours after being elected speaker of the House as the US government was in the 13th day of a partial shutdown over Trump's demand for $5 billion for a border wall.
A variety of strategies are being floated inside and outside the White House, among them trading wall funding for a deal on immigrants brought to the country as young people and now here illegally, or using a national emergency declaration to build the wall.
An effective legislator and political strategist, Pelosi played a key role in last year's midterm elections that saw Democrats seize control of the House for the first time in eight years by helping raise millions of dollars for Democratic candidates and keep them focus on the issue of health care on their campaign trails.
The pair also dismiss the idea of waiting for Robert Mueller to finish his investigation before beginning impeachment proceedings, and say it would be a betrayal of the American people to wait until the 2020 election to unseat him.
Nancy Pelosi (C) is surrounded by children after being elected the new speaker of the U.S. House of Representatives on Capitol Hill in Washington D.C., the United States, on January 3, 2019.
One day before Democrats take over the House of Representatives, Trump showed no signs of backing away from his insistence that Congress cough up $5.6 billion for his "beautiful" southern wall.
"The Democrats know they can't win based on all of the achievements of "Trump, ' so they are going all out on the desperately needed Wall and Border Security - and Presidential Harassment".
The U.S. House of Representatives passed a spending package late Thursday that would end the partial government shutdown amid a veto threat from the White House.
Congressional leaders were slated to meet with Trump at 11 a.m. Friday to resume negotiations, though neither side appears ready to cave.
The lower chamber on Thursday - the 13th day of the shutdown - passed six bills to fund the government for 2019 and an additional bill to fund the Department of Homeland Security through February 8.
Not content to merely imagine a world without Trump as president, Rep. Steve Cohen (D-TN) went after the governmental processes that gave the US Trump as president, proposing a constitutional amendment to eliminate the Electoral College. It was approved, 239-192.
The Democrats who control the House passed the bills that would fund the departments of State, Commerce, Agriculture, Labor, Treasury and other agencies through September 30, the end of the current fiscal year.
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The White House has rejected the Democratic package.
"He could not give a good answer", Schumer said.
The Homeland Security bill does not include Trump's border wall demands.
Democrats, on the other hand, spoke of families unable to pay bills and called on Trump to reopen the government while negotiations continue. Republicans have seized on the case to call for tougher border security.