Mr Trump Jr later said he had agreed to the meeting after being told he may get damaging material on Hillary Clinton.
President Trump has repeatedly denied any collusion with Russia.
The Post says some of the president's advisers fear the extent of the president's intervention could place him and some of his inner circle in legal jeopardy.
The Senate, the House of Representatives and a special counsel are all investigating alleged Russian interference in the presidential election - a claim denied by the Kremlin.
Legal concerns
The Washington Post first reported that President Trump had himself dictated the statement his son issued about the meeting with lawyer Natalia Veselnitskaya.
The newspaper cited multiple sources with knowledge of the deliberations.
ABC News later quoted two senior-level sources confirming this claim.
President Trump's advisers initially agreed that Mr Trump Jr would release a truthful statement that "couldn't be repudiated later if the full details emerged", the Washington Post says.
But it says that decision was reversed as President Trump was flying home from the G20 summit in Germany on 8 July.
In the statement, Mr Trump Jr said the meeting had "primarily discussed a programme about the adoption of Russian children", not campaign issues.
The statement was issued to the New York Times, as it was preparing its story on the meeting.
Mr Trump Jr later acknowledged that he had agreed to meet after being told Kremlin-linked information about Mrs Clinton would be offered during the talks.
He also released the email exchange that brought about the meeting.
Neither President Trump nor his son have commented on the latest media reports.
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