The claims are detailed in a letter by Elijah Cummings, top Democrat on the House oversight committee.
The White House fired Mr Flynn after he lied about meeting a Russian envoy.
He pleaded guilty last week to lying to FBI agents about his contacts with the Moscow diplomat.
It is alleged that while sitting a few yards away from the podium where President Trump was delivering his inaugural address on 20 January, Mr Flynn sent the texts to Alex Copson, managing partner of ACU Strategic Partners, a US nuclear power consultancy.
The White House aide was said to have been referring to plans by a consortium of US, Russian and French companies to build and operate 45 nuclear power plants in Saudi Arabia and other Arab countries.
However, a sticking point was US economic sanctions against Russia, which was to supply the reactors for the proposed project.
The outgoing Obama administration had just expanded sanctions on Russia to punish it for allegedly meddling in the 2016 US presidential election.
The Trump administration did not roll back the sanctions.
Congress later passed new measures against Russia, a bill that Mr Trump signed in August, though he said he did so reluctantly.
Mr Cummings detailed the Flynn claims in a letter to congressman Trey Gowdy, Republican chairman of the House Oversight and Government Reform Committee.
He called on Mr Gowdy to file legal orders to the White House, Mr Flynn and Mr Copson for documents on the nuclear power plan.
'Everything in place for us'
Mr Cummings wrote that Mr Flynn, a retired Army lieutenant general, had attempted to "manipulate the course of international nuclear policy for the financial gain of his business partners".
The congressman said he had heard about the alleged Flynn texts from an unnamed confidential informant.
Mr Cummings said the source was "authentic, credible, and reliable", and offered to introduce the individual to Mr Gowdy.
The congressman said the informant had met Mr Copson at an inaugural event in Washington DC on 20 January.
Mr Copson had allegedly shown the Flynn text message to the source, according to the congressman.
"Mike has been putting everything in place for us," Mr Copson allegedly told the whistleblower, Mr Cummings wrote.
"This is going to make a lot of very wealthy people."
The congressman wrote that the whistleblower was "extremely uncomfortable" with the conversation and left shortly thereafter.
'Certain investigative steps'
A lawyer for Mr Flynn, who was fired after only 24 days as Mr Trump's national security adviser, declined to comment.
President Trump's personal White House lawyer, Ty Cobb, also declined to comment.
ACU Strategic Partners has declined comment or make Mr Copson available for an interview.
Mr Flynn was a consultant to the firm for 14 months from 2015-16, according to amended financial disclosure forms he filed in August this year.
Mr Cummings wrote that he delayed releasing the letter at the request of Justice Department special counsel Robert Mueller until his team had "completed certain investigative steps".
Mr Mueller is investigating whether the Trump campaign colluded with Russia during the 2016 elections.
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