The annual speech would come during or soon after his trial in the Senate, where he is likely to be acquitted, the BBC reported on Friday.
He was formally impeached by the House of Representatives on Wednesday.
House Speaker Pelosi told a media outlet that he would be "impeached forever, no matter what the Senate does".
The impeachment vote in the Democrat-led House of Representatives split almost totally along party lines.
Democrats and Republicans have yet to agree on when the upcoming Senate trial will take place.
The letter sent by Pelosi on Friday begins with a pointed reminder to Trump of the "separation of powers" enshrined in the US Constitution.
The three branches - the judicial, executive, and legislative - she said are "co-equal branches acting as checks on each other".
"In the spirit of respecting our Constitution, I invite you to deliver your State of the Union Address," she wrote, adding: "Thank you for your attention to this matter".
White House spokesman Hogan Gidley said that Trump had accepted the invitation to speak.
A top Republican in congress, Texas Senator John Cornyn, reacted to the invitation with the hashtag "#somanymixedsignals".
"Guess she expects him to still be in office then," he tweeted.
Trump faces two charges, or articles of impeachment: abusing his power by requesting that Ukraine investigate his political rival so as to gain political advantage; and obstructing Congress by refusing to co-operate with the impeachment investigation.
The second charge came after Trump refused to allow White House officials to testify and withheld documents. Democrats charge that this breached constitutional protocol which requires Congress to provide oversight of the White House executive branch.
( With inputs from IANS )