WASHINGTON — President Trump denied again on Friday that there was any quid pro quo attached to his pressure on Ukraine to investigate his political enemies, but text messages and testimony collected by congressional investigators indicated that his own representatives saw it differently.

Envoys representing Mr. Trump sought to leverage the power of his office to prod Ukraine into opening investigations that would damage his Democratic opponents at home. They made clear to Ukrainian officials that the White House invitation their newly inaugurated president coveted depended on his commitment to the investigations.

And the senior American diplomat posted in Ukraine suspected it went even further than a trade of an Oval Office visit. He told colleagues that it appeared that unfreezing $391 million in American aid that Mr. Trump had blocked was contingent on the former Soviet republic following through on the politically charged investigations sought by the president and his personal lawyer Rudolph W. Giuliani, a conclusion sharply denied by another diplomat who said there were “no quid pro quo’s.”

The text messages, provided to three Democrat-led House committees by the former special envoy for Ukraine, Kurt D. Volker, may shape the impeachment inquiry now threatening the future of Mr. Trump’s presidency. They provided new pieces of a timeline of events in recent months and a road map for further investigation by House Democrats.