President Donald Trump vowed this morning to launch "a major investigation" into alleged voter fraud, providing little detail.

"I will be asking for a major investigation into VOTER FRAUD, including those registered to vote in two states, those who are illegal and ..." Trump tweeted today, adding in a second tweet, "even, those registered to vote who are dead (and many for a long time). Depending on results, we will strengthen up voting procedures!"

Trump offered no further information on what such an investigation would entail.

However, ABC News has learned that the White House has begun drafting a presidential memorandum ordering an investigation into voter fraud, according to a senior administration official.

Trump could sign the order as early as Thursday, after he meets with House Republicans in Philadelphia, the official said. The investigation could be conducted by either the Department of Homeland Security or the Department of Justice, and its primary focus would be on voter registration, not necessarily the extent of fraudulent votes cast in 2016, according to the official.

The president and members of his administration are repeating the unsubstantiated assertion that millions of people illegally cast votes in the 2016 presidential election -- a claim Trump first made after the election without giving evidence.

After winning the Electoral College but not the popular vote, Trump asserted in late November that there was "serious" voter fraud in three states. He suggested on Twitter that he would have won the popular vote "if you deduct the millions of people who voted illegally."

But state officials said there was no evidence to that effect, and the authors of studies that a Trump spokesman cited as evidence disputed the characterization.

White House press secretary Sean Spicer said Tuesday afternoon that Trump "does believe that" millions of illegal ballots were cast in the presidential election, citing "studies and information" that were presented to him as the basis of that belief. Spicer said at the time "maybe we will" investigate the alleged fraud.

The Public Interest Legal Foundation, an election integrity nonprofit based in Indiana, swiftly voiced its support for Trump's pledge to investigate alleged voter fraud, calling the move "important."

"The Obama administration had the tools to fight voter fraud but let them gather dust. Because of that neglect of their duties, aliens got on the rolls, people voted multiple times, and lawlessness took hold of our elections," said J. Christian Adams, the group's president and general counsel.

Ohio Secretary of State Jon Husted, a Republican, took to Twitter this morning to respond to Trump's tweets, saying, "We conducted a review 4 years ago in Ohio & already have a statewide review of 2016 election underway. Easy to vote, hard to cheat."

Speaking in 2013 about the state's review, Husted reportedly said, "Voter fraud does exist, but it is not an epidemic."

ABC News' Alexander Mallin, Lauren Pearle and Ryan Struyk contributed to this report.